Find the Derivative - d/dx y=cos(x^3)

Problem

d(cos(x^3))/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is cos(u) and the inner function is u=x^3

2. Apply the Chain Rule which states that (d(ƒ)*(g(x)))/d(x)=ƒ^′*(g(x))⋅g(x)^′

3. Differentiate the outer function with respect to the inner function. The derivative of cos(u) is −sin(u)

d(cos(x^3))/d(x)=−sin(x^3)⋅d(x^3)/d(x)

4. Differentiate the inner function using the Power Rule. The derivative of x^3 is 3*x^2

d(cos(x^3))/d(x)=−sin(x^3)⋅3*x^2

5. Simplify the expression by rearranging the terms.

d(cos(x^3))/d(x)=−3*x^2*sin(x^3)

Final Answer

d(cos(x^3))/d(x)=−3*x^2*sin(x^3)

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