Question

y = (5x)/(x ^ 5), Find * (dy)/(dx)​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:y = \dfrac{5x}{ {x}^{5} }$

can be rewritten as

$\rm :\longmapsto\:y = 5 {x}^{1 - 5}$

$\rm :\longmapsto\:y = 5 {x}^{ - 4}$

On differentiating both sides w. r. t. x, we get

$\rm :\longmapsto\:\dfrac{d}{dx}y =\dfrac{d}{dx} \: {5x}^{ - 4}$

We know,

$\boxed{ \bf{ \: \dfrac{d}{dx}k \: f(x) = k \: \dfrac{d}{dx}f(x)}}$

Using this, we get

$\rm :\longmapsto\:\dfrac{dy}{dx} =5 \: \dfrac{d}{dx} \: {x}^{ - 4}$

We know,

$\boxed{ \bf{ \: \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1}}}$

So, using this, we get

$\rm :\longmapsto\:\dfrac{dy}{dx} =5 \:( - 4) {x}^{ - 4 - 1}$

$\rm :\longmapsto\:\dfrac{dy}{dx} = - 20 {x}^{ - 5}$

$\bf :\longmapsto\:\dfrac{dy}{dx} = - \: \dfrac{20}{ {x}^{5} }$

Additional Information :-

$\boxed{ \bf{ \: \dfrac{d}{dx}k = 0}}$

$\boxed{ \bf{ \: \dfrac{d}{dx}x = 1}}$

$\boxed{ \bf{ \: \dfrac{d}{dx}logx = \frac{1}{x}}}$

$\boxed{ \bf{ \: \dfrac{d}{dx} {e}^{x} = {e}^{x} }}$

$\boxed{ \bf{ \: \dfrac{d}{dx} {a}^{x} = {a}^{x} \: loga}}$

$\boxed{ \bf{ \: \dfrac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x} }}}$