What is the derivative of cos(5x)?
Understand the Problem
The question is asking for the derivative of the function cos(5x). This involves applying the chain rule from calculus, where the derivative of cos(u) is -sin(u) multiplied by the derivative of u with respect to x.
Answer
-5sin(5x)
Answer for screen readers
The final answer is -5sin(5x)
Steps to Solve
- Identify the outer and inner functions
The outer function is $\cos(u)$ and the inner function is $u = 5x$.
- Differentiate the outer function
The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$.
- Differentiate the inner function
The derivative of $5x$ with respect to $x$ is $5$.
- Apply the chain rule
Multiply the derivative of the outer function by the derivative of the inner function: $$ \frac{d}{dx}\cos(5x) = -\sin(5x) \cdot 5 $$
Simplify the result: $$ \frac{d}{dx}\cos(5x) = -5\sin(5x) $$
The final answer is -5sin(5x)
More Information
The chain rule is a fundamental tool in calculus, used extensively for differentiating composite functions.
Tips
A common mistake is forgetting to multiply by the derivative of the inner function. Always remember to apply the chain rule correctly.