Find the Derivative - d/dx y=(8x^3+7)^(3/2)

Problem

d()/d(x)*(8*x3+7)(3/2)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule.

2. Apply the Power Rule to the outer function, which is u(3/2) where u=8*x3+7

3. Differentiate the inner function 8*x3+7 with respect to x

4. Multiply the derivative of the outer function by the derivative of the inner function.

d(y)/d(x)=3/2*(8*x3+7)(1/2)⋅d(8*x3+7)/d(x)

5. Calculate the derivative of the inner part using the Power Rule.

d(8*x3+7)/d(x)=24*x2

6. Simplify the resulting expression by multiplying the constants.

d(y)/d(x)=3/2⋅24*x2⋅(8*x3+7)(1/2)

d(y)/d(x)=36*x2*(8*x3+7)(1/2)

Final Answer

d(8*x3+7)/d(x)=36*x2√(,8*x3+7)

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