Find the Derivative - d/dx 1/27(9x^2+6)^(3/2)

Problem

d()/d(x)1/27*(9*x2+6)(3/2)

Solution

1. Identify the constant multiple rule, which allows the constant 1/27 to be moved outside the derivative.

1/27d(9*x2+6)/d(x)

2. Apply the power rule and the chain rule to the expression (9*x2+6)(3/2) by bringing down the exponent and subtracting one from it.

1/27⋅3/2*(9*x2+6)(1/2)⋅d(9*x2+6)/d(x)

3. Differentiate the inner function 9*x2+6 with respect to x

d(9*x2+6)/d(x)=18*x

4. Substitute the derivative of the inner function back into the expression.

1/27⋅3/2*(9*x2+6)(1/2)⋅18*x

5. Simplify the coefficients by multiplying 1/27 3/2 and 18

1/27⋅3/2⋅18=54/54=1

6. Combine the remaining terms to reach the final simplified derivative.

x*(9*x2+6)(1/2)

Final Answer

d()/d(x)1/27*(9*x2+6)(3/2)=x√(,9*x2+6)

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