Find the Derivative - d/dx ((x+5)/(x^2+2))^2

Problem

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

Solution

1. Apply the power rule and the chain rule by bringing the exponent 2 to the front and decreasing the power by 1

d()/d(x)*((x+5)/(x2+2))2=2*((x+5)/(x2+2))⋅d()/d(x)(x+5)/(x2+2)

2. Apply the quotient rule to differentiate the inner fraction u/v where u=x+5 and v=x2+2

d()/d(x)(x+5)/(x2+2)=((x2+2)⋅d(x+5)/d(x)−(x+5)⋅d(x2+2)/d(x))/((x2+2)2)

3. Evaluate the derivatives of the numerator and denominator terms.

d(x+5)/d(x)=1

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

4. Substitute the derivatives back into the quotient rule expression.

d()/d(x)(x+5)/(x2+2)=((x2+2)*(1)−(x+5)*(2*x))/((x2+2)2)

5. Simplify the numerator by distributing and combining like terms.

(x2+2)−(2*x2+10*x)=−x2−10*x+2

6. Combine all parts of the derivative and multiply the fractions.

d()/d(x)*((x+5)/(x2+2))2=2*((x+5)/(x2+2))⋅(−x2−10*x+2)/((x2+2)2)

7. Simplify the final expression by multiplying the numerators and denominators.

d()/d(x)*((x+5)/(x2+2))2=(2*(x+5)*(−x2−10*x+2))/((x2+2)3)

Final Answer

d()/d(x)*((x+5)/(x2+2))2=(2*(x+5)*(−x2−10*x+2))/((x2+2)3)

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