Question

e^3logx differentiate it

Answer 1

Answer:

e^3log x = e^log x^3 = x^3

So,

d(x^3)/dx = 3x^2

Answer 2

We have to find,

$\dfrac{d}{dx}\ e^{^{\displaystyle\small3\log(x)}}$

First let me convert the common logarithm to natural logarithm. We know it.

$\log(x)=\dfrac{\ln(x)}{\ln(10)}\ \ \ \left[\because\ \log_b(a)\ =\ \dfrac{\log_m(a)}{\log_m(b)}\right]$

So,

$\dfrac{d}{dx}\ e^{^{\displaystyle\small3\log(x)}}\ \implies\ \dfrac{d}{dx}\ e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}$

And we have the exponential function rule of differentiation.

$\dfrac{d}{dx}\ e^{f(x)}\ =\ e^{f(x)}\ \cdot\ \dfrac{d(f(x))}{dx}$

So,

$\dfrac{d}{dx}\ e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}\ \implies\ e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}\ \cdot\ \dfrac{d}{dx}\left(\dfrac{3\ln(x)}{\ln(10)}\right)\\ \\ \\ \implies\ e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}\ \cdot\ \dfrac{3}{\ln(10)}\ \cdot\ \dfrac{d}{dx}\ \ln(x)\\ \\ \\ \implies\ e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}\ \cdot \dfrac{3}{\ln(10)}\ \cdot\ \dfrac{1}{x}$

$\implies\ \boxed{\dfrac{3\cdot e^{\displaystyle\small\dfrac{3\ln(x)}{\ln(10)}}}{x\cdot \ln(10)}}$

Hence differentiated!