How do you find the derivative of f(x)= [(2x-5)^5]/[(x^2 +2)^2] using the chain rule?

Apr 18, 2018

$= \frac{10 {\left(2 x - 5\right)}^{4} \cdot {\left({x}^{2} + 2\right)}^{2} - {\left(2 x - 5\right)}^{5} \cdot 4 x \left({x}^{2} + 2\right)}{{x}^{2} + 2} ^ 4$

Explanation:

$f ' \left(x\right) = \frac{f ' \left(x\right) \cdot g \left(x\right) - f \left(x\right) \cdot g ' \left(x\right)}{g \left(x\right)} ^ 2$

$f ' \left(x\right) = \frac{\left(\left(5 {\left(2 x - 5\right)}^{4} \cdot 2\right) {\left({x}^{2} + 2\right)}^{2}\right) - {\left(2 x - 5\right)}^{5} \cdot \left(2 \left({x}^{2} + 2\right) \cdot 2 x\right)}{{\left({x}^{2} + 2\right)}^{2}} ^ 2$

$= \frac{10 {\left(2 x - 5\right)}^{4} \cdot {\left({x}^{2} + 2\right)}^{2} - {\left(2 x - 5\right)}^{5} \cdot 4 x \left({x}^{2} + 2\right)}{{x}^{2} + 2} ^ 4$

You can reduce more, but it's bored solve this equation, just use algebraic method.