Product and Chain rule derivative

Question

How to differentiate $\frac{2x}{(4x-3)^3}$ using the quotient rule?
How to differentiate $y= 2x(4x-3)^{-3}$ keeping it in powers form?

Answer 1

The quotient rule is given by $$ \frac{(d'u-u'd)}{d^2} $$ where $d'$ is the derivative of the denominator and $u'$ is the derivative of the numerator. Apply this formula and you will have your solution. If you are having trouble with this formula, please explain what you have tried.

If you are having trouble with the chain rule, the chain rule is given by $$ (f \circ g)'(x)=f'(g(x))g'(x). $$

For example, consider $f(x)=(2x+1)^4$ where $g(x)=2x+1$. Then, $$ g'(x) = 2.$$ Therefore, $$ f'(x) = 4(2x+1)^3 \times 2 = 8(2x+1)^3.$$

Answer 2

HINT

For the quotient let apply

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

and for the power

$$(f(x)h^n(x))'=f'(x)h^n(x)+f(x)\cdot nh^{n-1}(x)$$