Find the Derivative - d/dx y=(3x^2+5x+1)^(3/2)

Problem

d()/d(x)*(3*x2+5*x+1)(3/2)

Solution

1. Identify the outer function and the inner function to apply the chain rule. The outer function is u(3/2) and the inner function is u=3*x2+5*x+1

2. Apply the power rule to the outer function by bringing the exponent to the front and subtracting one from the exponent.

3/2*(3*x2+5*x+1)(1/2)

3. Differentiate the inner function with respect to x

d(3*x2+5*x+1)/d(x)=6*x+5

4. Multiply the derivative of the outer function by the derivative of the inner function according to the chain rule.

d(y)/d(x)=3/2*(3*x2+5*x+1)(1/2)*(6*x+5)

5. Simplify the expression by distributing the constant if possible or rearranging the terms.

d(y)/d(x)=(3*(6*x+5)√(,3*x2+5*x+1))/2

Final Answer

d()/d(x)*(3*x2+5*x+1)(3/2)=(3*(6*x+5)√(,3*x2+5*x+1))/2

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