Find the Derivative - d/dx a^9+cos(x)^9

Problem

d()/d(x)*(a9+cos9(x))

Solution

1. Identify the terms in the expression. The expression consists of a constant term a9 and a trigonometric power term cos9(x)

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

d()/d(x)*(a9+cos9(x))=d(a9)/d(x)+d(cos9(x))/d(x)

3. Differentiate the constant term. Since a is a constant, a9 is also a constant, and its derivative with respect to x is zero.

d(a9)/d(x)=0

4. Apply the power rule and the chain rule to the second term. For a function of the form u*(x)n the derivative is n⋅u*(x)(n−1)⋅d(u)/d(x) Here, u(x)=cos(x) and n=9

d(cos9(x))/d(x)=9*cos8(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+9*cos8(x)⋅(−sin(x))=−9*sin(x)*cos8(x)

Final Answer

d()/d(x)*(a9+cos9(x))=−9*sin(x)*cos8(x)

Want more problems? Check here!