# What is the derivative of #f(x) = ((x+5)/(x^2+2))^2#?

##
#f(x) = ((x+5)/(x^2+2))^2# ?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the derivative of ( f(x) = \left(\frac{x+5}{x^2+2}\right)^2 ), we use the chain rule and power rule.

Let ( u = \frac{x+5}{x^2+2} ). Then ( f(x) = u^2 ).

Now, differentiate ( u ) with respect to ( x ) to get ( \frac{du}{dx} ). Next, apply the power rule to ( u^2 ) and multiply by ( \frac{du}{dx} ).

First, find ( \frac{du}{dx} ): [ u = \frac{x+5}{x^2+2} ] [ \frac{du}{dx} = \frac{(x^2+2)(1) - (x+5)(2x)}{(x^2+2)^2} ]

Now, apply the power rule: [ f'(x) = 2u \frac{du}{dx} ] [ f'(x) = 2\left(\frac{x+5}{x^2+2}\right) \left(\frac{(x^2+2)(1) - (x+5)(2x)}{(x^2+2)^2}\right) ]

Simplify the expression to get the final derivative ( f'(x) ).

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7