Find the Derivative - d/dx x/(x^2+16)

Problem

d()/d(x)x/(x^2+16)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the variables for the numerator and denominator. Let u=x and v=x^2+16

3. Differentiate the individual components. The derivative of the numerator is d(x)/d(x)=1 and the derivative of the denominator is d(x^2+16)/d(x)=2*x

4. Substitute these values into the quotient rule formula.

d()/d(x)x/(x^2+16)=((x^2+16)*(1)−(x)*(2*x))/((x^2+16)^2)

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

(x^2+16−2*x^2)/((x^2+16)^2)

6. Combine the x^2 terms to reach the final simplified form.

(16−x^2)/((x^2+16)^2)

Final Answer

d()/d(x)x/(x^2+16)=(16−x^2)/((x^2+16)^2)

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