Sure, let's find the derivative of the function f(x) = cos³(5x).
We're going to use the Chain rule of differentiation, which states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is u³ (where u = cos(5x)) and the inner function is cos(5x).
Step 1: Find the derivative of the outer function:
Let's treat the whole of u = cos(5x) as if it is a single variable. Then, the outer function becomes u³. The derivative of u³ is 3u².
Step 2: Find the derivative of the inner function:
The inner function is cos(5x). The derivative of cos(x) is -sin(x), and the derivative of the inner function, applying chain rule again for 5x, is -5sin(5x).
Step 3: Apply the Chain Rule:
According to the Chain Rule, you multiply the derivatives of the outer and inner functions:
3u² * -5sin(5x)
Substitute u = cos(5x) back in to get:
-15sin(5x)*cos²(5x)
That's the derivative for this function.