Find the Derivative - d/dx -(10x)/((x^2+5)^2)

Problem

d()/d(x)*(−(10*x)/((x2+5)2))

Solution

1. Identify the constant and the rule needed. Pull the constant −10 out of the derivative and prepare to use the quotient rule, where u=x and v=(x2+5)2

−10⋅d()/d(x)x/((x2+5)2)

2. Apply the quotient rule formula (v⋅d(u)/d(x)−u⋅d(v)/d(x))/(v2)

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

3. Differentiate the numerator terms. Use the power rule for d(x)/d(x)=1 and the chain rule for d(x2+5)/d(x)=2*(x2+5)⋅2*x

−10⋅((x2+5)2⋅1−x⋅(2*(x2+5)⋅2*x))/((x2+5)4)

4. Simplify the numerator by factoring out the common term (x2+5)

−10⋅((x2+5)⋅((x2+5)−4*x2))/((x2+5)4)

5. Cancel the common factor (x2+5) from the numerator and denominator.

−10⋅(x2+5−4*x2)/((x2+5)3)

6. Combine like terms inside the numerator.

−10⋅(5−3*x2)/((x2+5)3)

7. Distribute the constant −10 to finalize the expression.

(−50+30*x2)/((x2+5)3)

Final Answer

d()/d(x)*(−(10*x)/((x2+5)2))=(30*x2−50)/((x2+5)3)

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