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

Problem

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

Solution

1. Identify the constant multiple rule, which allows the constant 2/3 to be moved outside the derivative.

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

2. Apply the power rule and the chain rule to the expression (x2+1)(3/2) by bringing the exponent down and subtracting one from it.

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

3. Differentiate the inner function x2+1 with respect to x

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

4. Combine the results and simplify the coefficients.

2/3⋅3/2*(x2+1)(1/2)⋅2*x

5. Simplify the expression by canceling the fractions and rearranging the terms.

1⋅(x2+1)(1/2)⋅2*x=2*x√(,x2+1)

Final Answer

d()/d(x)2/3*(x2+1)(3/2)=2*x√(,x2+1)

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