Derivative Notes

Published on: September 29, 2026

This Derivative Calculator differentiates a function and shows the working line by line. It first works out which rule the function is built from, because a sum, a product, a quotient and a composition each need a different move, and only then does it apply that rule the way the dedicated rule calculators on this site do. Nested pieces are differentiated inside their own steps. Following the same pattern every time makes it clear why a rule was chosen, not just what it produced.

Step-by-step method

Set up the problem.
Identify the main rule that applies.
Work through the steps of that rule.
Simplify and write the final derivative.

Formula:

\(\frac{d}{dx} f\left(x\right) = f'\left(x\right)\)

See Example 1 Hide Example 1

Problem

\(f\left(x\right) = 4 x^{3} - 2 x + 7\)

Approach

Step-by-step method

Set up the problem.
Identify the main rule that applies.
Work through the steps of that rule.
Simplify and write the final derivative.

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are given a difference. The separated terms are \(4 x^{3}\), \(2 x\), \(7\).

\(f\left(x\right) = 4 x^{3} - 2 x + 7\)

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) - g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) - \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the subtraction signs.

In this problem: Apply the derivative to each separated term, keeping each sign.

\(\frac{d}{dx}\left(4 x^{3} - 2 x + 7\right) = \frac{d}{dx}\left(4 x^{3}\right) - \frac{d}{dx}\left(2 x\right) + \frac{d}{dx}\left(7\right)\)

Step 4a

Step 4a - Differentiate the separated term \(4 x^{3}\).

In this problem: The separated term is \(4 x^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(4 x^{3}\right) = 12 x^{2}\)

Problem

\(f\left(x\right) = 4{x}^{3}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(4 x^{3}\). This is a constant multiple of a power of \(x\). The coefficient is \(4\), the power part is \(x^{3}\), and the exponent is \(n = 3\).

\(f\left(x\right) = 4{x}^{3}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(4\) is a constant coefficient.

\(\frac{d}{dx}4{x}^{3} = 4\cdot \frac{d}{dx}{x}^{3}\)

Problem

\(f\left(x\right) = 4 x^{3}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(4 x^{3}\). The constant coefficient is \(4\), and the variable part is \(x^{3}\).

\(4 x^{3} = 4\cdot \left(x^{3}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(4\) outside the derivative and leave \(x^{3}\) inside the derivative.

\(\frac{d}{dx}\left(4 x^{3}\right) = 4\cdot \frac{d}{dx}\left(x^{3}\right)\)

Final Answer

\(4\cdot \frac{d}{dx}\left(x^{3}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 3\). Substitute this exponent into the power rule, then keep the coefficient \(4\) outside.

\(4\cdot \frac{d}{dx}{x}^{3} = 4\cdot 3{x}^{3 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{3}\), multiply by the constant \(4\), and keep the solving line in power form. This gives \(12{x}^{2}\).

\(4\cdot 3{x}^{2} = 12{x}^{2}\)

Final Answer

\(12 x^{2}\)

Step 4b

Step 4b - Differentiate the separated term \(2 x\).

In this problem: The separated term is \(2 x\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(2 x\right) = 2\)

Problem

\(f\left(x\right) = 2 x\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(x\right) = 2x \Leftrightarrow 2{x}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(2\) is a constant coefficient.

\(\frac{d}{dx}2{x}^{1} = 2\cdot \frac{d}{dx}{x}^{1}\)

Problem

\(f\left(x\right) = 2 x^{1}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(2 x^{1}\). The constant coefficient is \(2\), and the variable part is \(x^{1}\).

\(2 x^{1} = 2\cdot \left(x^{1}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(2\) outside the derivative and leave \(x^{1}\) inside the derivative.

\(\frac{d}{dx}\left(2 x^{1}\right) = 2\cdot \frac{d}{dx}\left(x^{1}\right)\)

Final Answer

\(2\cdot \frac{d}{dx}\left(x^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(2\) outside.

\(2\cdot \frac{d}{dx}{x}^{1} = 2\cdot {x}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(2\).

\(2\cdot {x}^{0} = 2\cdot 1 = 2\)

Final Answer

\(2\)

Step 4c

Step 4c - Differentiate the separated term \(7\).

In this problem: The separated term is \(7\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(7\right) = 0\)

Problem

\(f\left(x\right) = 7\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(7\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = 7\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(7\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(7\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, keeping the signs, then simplify.

\(f'\left(x\right) = 12 x^{2} - 2 + 0 = 12 x^{2} - 2\)

Final Answer

\(f'\left(x\right) = 12 x^{2} - 2\)

See Example 2 Hide Example 2

Problem

\(f\left(x\right) = x^{2} \cos{\left(x \right)}\)

Approach

Step-by-step method

Set up the problem.
Identify the main rule that applies.
Work through the steps of that rule.
Simplify and write the final derivative.

Step 1

Step 1 - Set up the two factors f(x) and g(x).

In this problem: The first factor is \(x^{2}\) and the second factor is \(\cos{\left(x \right)}\).

\(\begin{gathered} f\left(x\right) = x^{2} \\ g\left(x\right) = \cos{\left(x \right)} \end{gathered}\)

Step 2

Step 2 - Write the product rule formula.

In this problem: The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.

\(\frac{d}{dx}\left(f\left(x\right) \cdot g\left(x\right)\right) = f'\left(x\right)g\left(x\right) + f\left(x\right)g'\left(x\right)\)

Step 3

Step 3 - Differentiate the first factor.

In this problem: Differentiate \(x^{2}\) to get \(2 x\).

\(f'\left(x\right) = \frac{d}{dx}\left(x^{2}\right) = 2 x\)

Step 4

Step 4 - Differentiate the second factor.

In this problem: Differentiate \(\cos{\left(x \right)}\) to get \(- \sin{\left(x \right)}\).

\(g'\left(x\right) = \frac{d}{dx}\left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}\)

Step 5

Step 5 - Substitute into the product rule formula.

In this problem: Substitute the factors and their derivatives into the formula.

\(y'\left(x\right) = \left(2 x\right) \cdot \left(\cos{\left(x \right)}\right) + \left(x^{2}\right) \cdot \left(- \sin{\left(x \right)}\right)\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the products and combine like terms.

\(y'\left(x\right) = - x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = - x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)}\)

See Example 3 Hide Example 3

Problem

\(f\left(x\right) = 2 x^{3} + 5 x^{2} - 4\)

Approach

Step-by-step method

Set up the problem.
Identify the main rule that applies.
Work through the steps of that rule.
Simplify and write the final derivative.

Step 1

Step 1 - Set up the terms in the difference.

In this problem: We are given a difference. The separated terms are \(2 x^{3}\), \(5 x^{2}\), \(4\).

\(f\left(x\right) = 2 x^{3} + 5 x^{2} - 4\)

Step 2

Step 2 - Write the difference rule formula.

In this problem: The difference rule says the derivative of a difference is the difference of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) - g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) - \frac{d}{dx}g\left(x\right)\)

Step 3

Step 3 - Split the derivative across the subtraction signs.

In this problem: Apply the derivative to each separated term, keeping each sign.

\(\frac{d}{dx}\left(2 x^{3} + 5 x^{2} - 4\right) = \frac{d}{dx}\left(2 x^{3}\right) + \frac{d}{dx}\left(5 x^{2}\right) - \frac{d}{dx}\left(4\right)\)

Step 4a

Step 4a - Differentiate the separated term \(2 x^{3}\).

In this problem: The separated term is \(2 x^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(2 x^{3}\right) = 6 x^{2}\)

Problem

\(f\left(x\right) = 2{x}^{3}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(2 x^{3}\). This is a constant multiple of a power of \(x\). The coefficient is \(2\), the power part is \(x^{3}\), and the exponent is \(n = 3\).

\(f\left(x\right) = 2{x}^{3}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(2\) is a constant coefficient.

\(\frac{d}{dx}2{x}^{3} = 2\cdot \frac{d}{dx}{x}^{3}\)

Problem

\(f\left(x\right) = 2 x^{3}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(2 x^{3}\). The constant coefficient is \(2\), and the variable part is \(x^{3}\).

\(2 x^{3} = 2\cdot \left(x^{3}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(2\) outside the derivative and leave \(x^{3}\) inside the derivative.

\(\frac{d}{dx}\left(2 x^{3}\right) = 2\cdot \frac{d}{dx}\left(x^{3}\right)\)

Final Answer

\(2\cdot \frac{d}{dx}\left(x^{3}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 3\). Substitute this exponent into the power rule, then keep the coefficient \(2\) outside.

\(2\cdot \frac{d}{dx}{x}^{3} = 2\cdot 3{x}^{3 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{3}\), multiply by the constant \(2\), and keep the solving line in power form. This gives \(6{x}^{2}\).

\(2\cdot 3{x}^{2} = 6{x}^{2}\)

Final Answer

\(6 x^{2}\)

Step 4b

Step 4b - Differentiate the separated term \(5 x^{2}\).

In this problem: The separated term is \(5 x^{2}\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(5 x^{2}\right) = 10 x\)

Problem

\(f\left(x\right) = 5{x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(5 x^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(5\), the power part is \(x^{2}\), and the exponent is \(n = 2\).

\(f\left(x\right) = 5{x}^{2}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(5\) is a constant coefficient.

\(\frac{d}{dx}5{x}^{2} = 5\cdot \frac{d}{dx}{x}^{2}\)

Problem

\(f\left(x\right) = 5 x^{2}\)

Approach

Step-by-step method

Set up the problem.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(5 x^{2}\). The constant coefficient is \(5\), and the variable part is \(x^{2}\).

\(5 x^{2} = 5\cdot \left(x^{2}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(5\) outside the derivative and leave \(x^{2}\) inside the derivative.

\(\frac{d}{dx}\left(5 x^{2}\right) = 5\cdot \frac{d}{dx}\left(x^{2}\right)\)

Final Answer

\(5\cdot \frac{d}{dx}\left(x^{2}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(5\) outside.

\(5\cdot \frac{d}{dx}{x}^{2} = 5\cdot 2{x}^{2 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{2}\), multiply by the constant \(5\), and keep the solving line in power form. This gives \(10{x}^{1}\).

\(5\cdot 2{x}^{1} = 10{x}^{1} = 10 x\)

Final Answer

\(10 x\)

Step 4c

Step 4c - Differentiate the separated term \(4\).

In this problem: The separated term is \(4\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(4\right) = 0\)

Problem

\(f\left(x\right) = 4\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(4\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = 4\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(4\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(4\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

In this problem: Combine the derivative from each separated term, keeping the signs, then simplify.

\(f'\left(x\right) = 6 x^{2} + 10 x - 0 = 6 x^{2} + 10 x\)

Final Answer

\(f'\left(x\right) = 6 x^{2} + 10 x\)

See Example 4 Hide Example 4

Problem

\(f\left(x\right) = x^{3} \sin{\left(x \right)}\)

Approach

Step-by-step method

Set up the problem.
Identify the main rule that applies.
Work through the steps of that rule.
Simplify and write the final derivative.

Step 1

Step 1 - Set up the two factors f(x) and g(x).

In this problem: The first factor is \(x^{3}\) and the second factor is \(\sin{\left(x \right)}\).

\(\begin{gathered} f\left(x\right) = x^{3} \\ g\left(x\right) = \sin{\left(x \right)} \end{gathered}\)

Step 2

Step 2 - Write the product rule formula.

In this problem: The derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second.

\(\frac{d}{dx}\left(f\left(x\right) \cdot g\left(x\right)\right) = f'\left(x\right)g\left(x\right) + f\left(x\right)g'\left(x\right)\)

Step 3

Step 3 - Differentiate the first factor.

In this problem: Differentiate \(x^{3}\) to get \(3 x^{2}\).

\(f'\left(x\right) = \frac{d}{dx}\left(x^{3}\right) = 3 x^{2}\)

Step 4

Step 4 - Differentiate the second factor.

In this problem: Differentiate \(\sin{\left(x \right)}\) to get \(\cos{\left(x \right)}\).

\(g'\left(x\right) = \frac{d}{dx}\left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}\)

Step 5

Step 5 - Substitute into the product rule formula.

In this problem: Substitute the factors and their derivatives into the formula.

\(y'\left(x\right) = \left(3 x^{2}\right) \cdot \left(\sin{\left(x \right)}\right) + \left(x^{3}\right) \cdot \left(\cos{\left(x \right)}\right)\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the products and combine like terms.

\(y'\left(x\right) = x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}\)