Find the Derivative - d/d@VAR f(x)=(x^3-8)^(2/3)

Problem

d()/d(x)*(x3−8)(2/3)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is u(2/3) and the inner function is u=x3−8

2. Differentiate the outer function with respect to u using the Power Rule.

d()/d(u)*u(2/3)=2/3*u(−1/3)

3. Differentiate the inner function with respect to x

d()/d(x)*(x3−8)=3*x2

4. Apply the Chain Rule by multiplying the derivative of the outer function by the derivative of the inner function.

d(ƒ)/d(x)=2/3*(x3−8)(−1/3)⋅3*x2

5. Simplify the expression by canceling the constant factors and rewriting the negative exponent.

d(ƒ)/d(x)=2*x2*(x3−8)(−1/3)

Final Answer

d()/d(x)*(x3−8)(2/3)=(2*x2)/√(3,x3−8)

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