Trigonometric Derivative Notes

Published on: September 29, 2026

This Trigonometric Derivative Calculator differentiates sine, cosine, tangent and their relatives, and shows the working line by line. Each function has its own fixed derivative, and the ones built on cosine carry a minus sign that the others do not. A constant multiple in front is carried through unchanged. Following the same pattern every time keeps the signs attached to the right functions.

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\begin{gathered} \frac{d}{dx}\left(\sin{\left(x \right)}\right) = \cos{\left(x \right)} \\ \frac{d}{dx}\left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)} \\ \frac{d}{dx}\left(\tan{\left(x \right)}\right) = \sec^{2}{\left(x \right)} \end{gathered}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the trigonometric function and any constant multiple.

In this problem: The function is \(\sin{\left(x \right)}\).

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

Step 2

Step 2 - Write the derivative rule for that trigonometric function.

In this problem: The derivative of \(\sin{\left(x \right)}\) is \(\cos{\left(x \right)}\).

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

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = \cos{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = \cos{\left(x \right)}\)
See Example 2 Hide Example 2

Problem

\(f\left(x\right) = 5 \cos{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the trigonometric function and any constant multiple.

In this problem: The function is \(\cos{\left(x \right)}\). The constant multiple \(5\) stays in front.

\(f\left(x\right) = 5 \cos{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that trigonometric function.

In this problem: The derivative of \(\cos{\left(x \right)}\) is \(- \sin{\left(x \right)}\).

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

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = 5 \cdot \left(- \sin{\left(x \right)}\right) = - 5 \sin{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = - 5 \sin{\left(x \right)}\)
See Example 3 Hide Example 3

Problem

\(f\left(x\right) = 4 \sin{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the trigonometric function and any constant multiple.

In this problem: The function is \(\sin{\left(x \right)}\). The constant multiple \(4\) stays in front.

\(f\left(x\right) = 4 \sin{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that trigonometric function.

In this problem: The derivative of \(\sin{\left(x \right)}\) is \(\cos{\left(x \right)}\).

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

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = 4 \cdot \left(\cos{\left(x \right)}\right) = 4 \cos{\left(x \right)}\)

Final Answer

\(f'\left(x\right) = 4 \cos{\left(x \right)}\)
See Example 4 Hide Example 4

Problem

\(f\left(x\right) = 3 \cos{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the trigonometric function and any constant multiple.

In this problem: The function is \(\cos{\left(x \right)}\). The constant multiple \(3\) stays in front.

\(f\left(x\right) = 3 \cos{\left(x \right)}\)

Step 2

Step 2 - Write the derivative rule for that trigonometric function.

In this problem: The derivative of \(\cos{\left(x \right)}\) is \(- \sin{\left(x \right)}\).

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

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the function with its derivative and simplify.

\(f'\left(x\right) = 3 \cdot \left(- \sin{\left(x \right)}\right) = - 3 \sin{\left(x \right)}\)

Final Answer

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