Find the Derivative - d/dx y=a^8+cos(x)^8

Problem

d()/d(x)*(a8+cos8(x))

Solution

1. Identify the terms in the expression. The expression consists of a constant term a8 and a power of a trigonometric function cos8(x)

2. Apply the sum rule for derivatives. The derivative of a sum is the sum of the derivatives of each term.

3. Differentiate the constant term. Since a is a constant, a8 is also a constant, and its derivative with respect to x is zero.

d(a8)/d(x)=0

4. Apply the power rule and the chain rule to the second term. For a function in the form un the derivative is n*u(n−1)⋅d(u)/d(x) Here, u=cos(x) and n=8

d(cos8(x))/d(x)=8*cos7(x)⋅d(cos(x))/d(x)

5. Differentiate the inner function. The derivative of cos(x) is −sin(x)

d(cos(x))/d(x)=−sin(x)

6. Combine the results and simplify the expression.

0+8*cos7(x)⋅(−sin(x))=−8*cos7(x)*sin(x)

Final Answer

d()/d(x)*(a8+cos8(x))=−8*cos7(x)*sin(x)

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