Find the Derivative - d/dx 1/3(x^2+2)^(3/2)

Problem

d()/d(x)1/3*(x2+2)(3/2)

Solution

1. Identify the constant multiple rule, which allows the constant 1/3 to be moved outside the derivative.

1/3d(x2+2)/d(x)

2. Apply the power rule and the chain rule to the expression (x2+2)(3/2) where the outer function is u(3/2) and the inner function is u=x2+2

1/3⋅3/2*(x2+2)(3/2−1)⋅d(x2+2)/d(x)

3. Simplify the coefficients by multiplying 1/3 and 3/2 and subtract the exponents.

1/2*(x2+2)(1/2)⋅d(x2+2)/d(x)

4. Differentiate the inner function x2+2 with respect to x

1/2*(x2+2)(1/2)⋅2*x

5. Simplify the final expression by canceling the 1/2 and 2

x*(x2+2)(1/2)

Final Answer

d()/d(x)1/3*(x2+2)(3/2)=x√(,x2+2)

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