Question

find d/dx when y=log e(x^3)​

Answer 1

Answer :

dy/dx=3/x

Step by step explanatìon:

Given :

$\sf\:y=\log_e(x^3)$

To Find :

dy /dx

Formula's used :

$\sf\dfrac{d(\log_{e}x)}{dx}=\dfrac{1}{x}$

$\sf\dfrac{d(constant)}{dx}=0$

Solution :

We have :

$\sf\:y=\log_e(x^3)$

Now ,Differentiate with respect to x,by chain rule

$\sf\dfrac{dy}{dx}=\dfrac{d(\log_{e}x^3)}{d(x^3)}\times\dfrac{d(x^3)}{dx}$

$\sf\dfrac{dy}{dx}=\dfrac{1}{x^3}\times\:3x^2$

$\sf\dfrac{dy}{dx}=\dfrac{3x^2}{x^3}$

$\implies\sf\dfrac{dy}{dx}=\dfrac{3}{x}$

More About the topic :

•Chain Rule

Let y=f(t) ,t = g(u) and u =m(x) ,then

$\sf\:\dfrac{dy}{dx}=\dfrac{dy}{dt}\times \dfrac{dt}{du}\times\dfrac{du}{dx}$

• Differention Formula's

$1)\sf\frac{d(\cos\:x)}{dx}=-\sin\:x$

$2)\sf\frac{d(\sin\:x)}{dx}=\cos\:x$

$3)\sf\frac{d(e^x)}{dx}=e^x$

$4)\sf\frac{d(\tan\:x)}{dx}=\sec^2\:x$

•If $\sf\:y=\log_{a}x$

Then,

$\bf\dfrac{dy}{dx}=\dfrac{1}{(\log\:a)x}$

Answer 2

Given ,

The function is

• $\tt y = log_{e} {(x)}^{3}$

Differentiating y wrt x by chain rule , we get

$\tt \implies \frac{dy}{dx} = \frac{d \{log_{e} {(x)}^{3} \} }{dx}$

$\tt \implies \frac{dy}{dx} = \frac{1}{ {(x)}^{3} } . \frac{d {(x)}^{3} }{dx}$

$\tt \implies \frac{dy}{dx} = \frac{1}{ {(x)}^{3} } .3 {(x)}^{2}$

$\tt \implies \frac{dy}{dx} = \frac{3}{x}$

Second Method :

We know that ,

$\boxed{ \tt{ log_{e}{(x)}^{n}= n.log_{e}(x) }}$

So ,

$\tt y = log_{e} {(x)}^{3} = 3.log_{e} (x)$

Now , differentiating wrt x , we get

$\tt \implies \frac{dy}{dx} = \frac{d \{3. log_{e}(x) \} }{dx}$

$\tt \implies \frac{dy}{dx} = log_{e}(x) \frac{d(3)}{dx} + 3 \frac{d \{ log_{e}(x) \} }{dx} \: \: \: \{ \because \: product \: rule \}$

$\tt \implies \frac{dy}{dx} = \frac{3}{x}$