Find the Derivative - d/dx x/(x^2+81)

Problem

d()/d(x)x/(x2+81)

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=x and v=x2+81

3. Differentiate the individual components. Find d(x)/d(x)=1 and d(x2+81)/d(x)=2*x

4. Substitute these values into the quotient rule formula.

d()/d(x)x/(x2+81)=((x2+81)*(1)−(x)*(2*x))/((x2+81)2)

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

(x2+81−2*x2)/((x2+81)2)

6. Combine the x2 terms to reach the final simplified form.

(81−x2)/((x2+81)2)

Final Answer

d()/d(x)x/(x2+81)=(81−x2)/((x2+81)2)

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