Trigonometric Derivative Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Trigonometric Derivative Calculator helps you differentiate the six trigonometric functions such as sin(x), 5cos(x), or tan(x). Each trigonometric function has its own derivative rule, and any constant multiple simply stays in front.

Step-by-step method

  1. Identify the trigonometric function and any constant multiple.
  2. Write the derivative rule for that trigonometric function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

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

Example 1:

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

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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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:

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