Derivative Calculator

Published on: April 6, 2025

Final Answer: Free • Full Steps: Plus

This Derivative Calculator differentiates a function step by step. It first identifies which derivative rule fits your input - constant, variable, constant multiple, power, sum, difference, product, quotient, chain, trigonometric, logarithmic, or exponential - and then works through that rule's own method, the same way the dedicated rule calculators on this site do.

Step-by-step method

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

Formula:

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

Example 1: solved with the difference rule.

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

Step 1 - Set up the terms in the difference.

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

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

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 - Split the derivative across the subtraction signs.

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

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

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

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

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

Problem

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

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 \(3 x^{2}\). This is a constant multiple of a power of \(x\). The coefficient is \(3\), the power part is \(x^{2}\), and the exponent is \(n = 2\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(3\) is a constant coefficient.

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

Problem

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

Approach

Step-by-step method

Set up the coefficient and the variable part.
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 \(3 x^{2}\). The constant coefficient is \(3\), and the variable part is \(x^{2}\).

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

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 \(3\) outside the derivative and leave \(x^{2}\) inside the derivative.

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

Final Answer

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

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 \(3\) outside.

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

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{2}\), multiply by the constant \(3\), and keep the solving line in power form. This gives \(6{x}^{1}\).

\(3\cdot 2{x}^{1} = 6{x}^{1} = 6 x\)

Final Answer

\(6 x\)

Step 4b - Differentiate the separated term \(5 x\).

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

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

Problem

\(f\left(x\right) = 5 x\)

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 \(5\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(x\right) = 5x \Leftrightarrow 5{x}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

\(f\left(x\right) = 5 x^{1}\)

Approach

Step-by-step method

Set up the coefficient and the variable part.
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 \(5 x^{1}\). The constant coefficient is \(5\), and the variable part is \(x^{1}\).

\(5 x^{1} = 5\cdot \left(x^{1}\right)\)

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 \(5\) outside the derivative and leave \(x^{1}\) inside the derivative.

\(\frac{d}{dx}\left(5 x^{1}\right) = 5\cdot \frac{d}{dx}\left(x^{1}\right)\)

Final Answer

\(5\cdot \frac{d}{dx}\left(x^{1}\right)\)

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 \(5\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

\(5\cdot {x}^{0} = 5\cdot 1 = 5\)

Final Answer

\(5\)

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 - Combine and simplify.

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

\(f'\left(x\right) = 6 x + 5 - 0 = 6 x + 5\)

Final answer:

\(f'\left(x\right) = 6 x + 5\)

Example 2: solved with the product rule.

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

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 \(\sin{\left(x \right)}\).

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

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 - 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 - 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 - Substitute into the product rule formula.

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

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

Step 6 - Simplify the result.

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

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

Final answer:

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