Find the Derivative - d/dx cos(x)^8

Problem

d(cos(x))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The expression is of the form un where u=cos(x) and n=8

2. Apply the Power Rule to the outer function by bringing the exponent to the front and subtracting one from the exponent.

d(u8)/d(u)=8*u7

3. Differentiate the inner function u=cos(x) with respect to x

d(cos(x))/d(x)=−sin(x)

4. Combine the results using the Chain Rule formula d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

8*cos(x)⋅(−sin(x))

5. Simplify the expression by moving the negative sign to the front.

−8*cos(x)*sin(x)

Final Answer

d(cos(x))/d(x)=−8*cos(x)*sin(x)

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