Question

Differentiate \(\rm e^{1+x^2}\) with respect to ln x
1. \(\rm 2x^2e^{1+x^2}\)
2. \(\rm xe^{1+x^2}\)
3. \(\rm 2xe^{1+x^2}\)
4. \(\rm x^2e^{1+x^2}\)

Answer 1

Correct Answer - Option 1 : \(\rm 2x^2e^{1+x^2}\)

Concept:

​Parametric Form:

If f(x) and g(x) are the functions in x, then 

\(\rm df(x)\over dg(x)\) = \(\rm \frac{df(x)\over dx}{dg(x)\over dx}\) 

Calculation:

Let f(x) = \(\rm e^{1+x^2}\) and g(x) = ln x

\(\rm \frac{df(x)}{dx}\) = \(\rm e^{1+x^2}\)(2x)

\(\rm \frac{df(x)}{dx}\) = 2x \(\rm e^{1+x^2}\)

Also

\(\rm \frac{dg(x)}{dx}\) = \(\rm 1\over x\)

Now Differentiation of f(x) with respect to g(x) is 

\(\rm df(x)\over dg(x)\) = \(\rm \frac{df(x)\over dx}{dg(x)\over dx}\) 

\(\rm df(x)\over dg(x)\) = \(\rm \frac{2xe^{1+x^2}}{1\over x}\)

\(\rm df(x)\over dg(x)\) = 2x²\(\boldsymbol{\rm e^{1+x^2}}\)