Calculators / Calculus 1 / Derivative Calculator

Derivative Calculator

Published on: March 23, 2025

This Derivative Calculator helps you find the derivative of a function and shows the working clearly step by step. It can be used to differentiate common algebraic expressions and understand how rules such as the power rule, sum rule, and other basic differentiation rules are applied. This makes it useful for checking answers, learning derivative methods, and practising calculus more confidently.

Step-by-step method

Set up the problem.
Pick the rule.
Apply the rule.
Simplify.

Formula:

c = 0

x = 1

xn = n·xn−1

k·f(x) = k·f'(x)

f(x)+g(x) = f'(x)+g'(x)

f(x)−g(x) = f'(x)−g'(x)

f(x)·g(x) = f'(x)g(x)+f(x)g'(x)

f(x)/g(x) =

f'(x)g(x) − f(x)g'(x)

g(x)2

f(g(x)) = f'(g(x))·g'(x)

ex = ex

ax = ax·ln(a)

ln(x) =

log_a(x) =

x·ln(a)

sin(x) = cos(x)

cos(x) = −sin(x)

tan(x) = sec(x)2

sec(x) = sec(x)·tan(x)

csc(x) = −csc(x)·cot(x)

cot(x) = −csc(x)2

arcsin(x) =

√(1 − x2)

arccos(x) =

−1

√(1 − x2)

arctan(x) =

1 + x2

Example 1: f(x) = 3x^2+5x-4

Step 1 - Set up the problem.

In this problem: We will differentiate f(x) with respect to x.

f(x)=3x2 + 5x − 4

Step 2 - Pick the rule.

In this problem: Use the Sum/Difference rule: differentiate each term separately (shown in one line).

(3x2 + 5x − 4) ⇒

(3x2) +

(5x) −

(4)

Step 3A - Set up this term.

In this problem: Use the Constant multiple rule: pull out the constant. Here k = 3 and u = x^2.

(3x2) = 3·

(x2)

Step 3B - Pick the rule (identity).

In this problem: Now differentiate the inside using the Power rule. Here u = x and n = 2.

xn = n·xn−1

Step 3C - Apply the rule.

In this problem: Compute the inside derivative, then multiply by k = 3.

(3x2) = 3·(2x) = 6x

Step 3D - Set up this term.

In this problem: Use the Constant multiple rule: pull out the constant. Here k = 5 and u = x.

(5x) = 5·

(x)

Step 3E - Pick the rule (identity).

In this problem: Now differentiate the inside using the Derivative of x. Here the inside is x.

x = 1

Step 3F - Apply the rule.

In this problem: Compute the inside derivative, then multiply by k = 5.

(5x) = 5·(1) = 5

Step 3G - Set up this term.

In this problem: Write the derivative operator for this term.

(4)

Step 3H - Pick the rule (identity).

In this problem: Use the Constant rule. Here c = 4.

c = 0

Step 3I - Apply the rule.

In this problem: Apply the identity to get the derivative of this term.

(4) = 0

Step 3J - Combine the results.

In this problem: Add the differentiated terms back together to form f'(x). No further simplification is needed.

f'(x)=6x + 5

Final answer: f'(x) = 6x + 5

Example 2: f(x) = (1/2)x^3+sin(x)

Step 1 - Set up the problem.

In this problem: We will differentiate f(x) with respect to x.

f(x)=

x3 + sin(x)

Step 2 - Pick the rule.

In this problem: Use the Sum/Difference rule: differentiate each term separately (shown in one line).

(

x3 + sin(x)) ⇒

(

x3) +

(sin(x))

Step 3A - Set up this term.

In this problem: Use the Constant multiple rule: pull out the constant. Here k = 1/2 and u = x^3.

(

x3) =

(x3)

Step 3B - Pick the rule (identity).

In this problem: Now differentiate the inside using the Power rule. Here u = x and n = 3.

xn = n·xn−1

Step 3C - Apply the rule.

In this problem: Compute the inside derivative, then multiply by k = 1/2.

(

x3) =

·(3x2) =

Step 3D - Set up this term.

In this problem: Write the derivative operator for this term.

(sin(x))

Step 3E - Pick the rule (identity).

In this problem: Use the Trig rule (sin). Here u = x.

sin(x) = cos(x)

Step 3F - Apply the rule.

In this problem: Apply the identity to get the derivative of this term.

(sin(x)) = cos(x)

Step 3G - Combine the results.

In this problem: Add the differentiated terms back together to form f'(x). No further simplification is needed.

f'(x)=

x2 + cos(x)

Final answer: f'(x) = 3x^2/2 + cos(x)