Find the Derivative - d/d@VAR f(x)=x/(x^2+25)

Problem

d()/d(x)x/(x2+25)

Solution

1. Identify the function as a quotient of two terms, u=x and v=x2+25 which requires the quotient rule.

2. Recall the quotient rule formula for a function u/v

d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=x and the denominator v=x2+25 individually.

d(x)/d(x)=1

d(x2+25)/d(x)=2*x

4. Substitute these components into the quotient rule formula.

d()/d(x)x/(x2+25)=((x2+25)*(1)−(x)*(2*x))/((x2+25)2)

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

(x2+25−2*x2)/((x2+25)2)

(25−x2)/((x2+25)2)

Final Answer

d()/d(x)x/(x2+25)=(25−x2)/((x2+25)2)

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