Derivative of e^x^2 by First Principle and Chain Rule

The function e to the power x² is written as $e^{x^2}$ and its derivative is $2x{e}^{x^2}$. In this post, we will find the derivative of e to the power x square by the first principle and chain rule of derivatives.

Recall the first principle of derivatives: The derivative of a function f(x) by first principle is defined by the limit:

${\displaystyle \frac{d}{dx}}(f(x))$ = lim_h→0 ${\displaystyle \frac{f(x+h)-f(x)}{h}}$ $\cdots$ (I)

We will use this to find the differentiation of e to the power x².

Derivative of $e^{x^2}$ by First Principle

Let us put $f(x)=e^{x^2}$ in the above formula (I).

So we obtain that

${\displaystyle \frac{d}{dx}}(e^{x^2})$ = lim_h→0 ${\displaystyle \frac{e^{(x+h{)}^2}-e^{x^2}}{h}}$

= lim_h→0 ${\displaystyle \frac{e^{x^2+2xh+h^2}-e^{x^2}}{h}}$. Here we have used the formula (a+b)² = a²+2ab+b².

= lim_h→0 ${\displaystyle \frac{e^{x^2}{e}^{2xh+h^2}-e^{x^2}}{h}}$

= lim_h→0 ${\displaystyle \frac{e^{x^2}(e^{2xh+h^2}-1)}{h}}$

= lim_h→0 ${\displaystyle \frac{e^{x^2}(e^{h(2x+h)}-1)}{h}}$

= lim_h→0 $[{\displaystyle \frac{e^{x^2}(e^{h(2x+h)}-1)}{h(2x+h)}}$ $\times (2x+h)]$

$=e^{x^2}\underset{h\to 0}{lim}$ ${\displaystyle \frac{e^{h(2x+h)}-1}{h(2x+h)}}$ $\times \underset{h\to 0}{lim}(2x+h)$

Let u=h(2x+h). Then u tends to zero when h tends to 0.

$=e^{x^2}\underset{u\to 0}{lim}$ ${\displaystyle \frac{e^u-1}{u}}$ $\times \underset{h\to 0}{lim}(2x+h)$

$=e^{x^2}\times 1$ $\times (2x+0)$ as the limit of (e^x-1)/x is 1 when x tends to 0.

$=2x{e}^{x^2}$

Thus, the derivative of e^x^2 by first principle is 2xe^x^2.

Derivative of $e^{x^2}$ by Chain Rule

Let u=x².

Then ${\displaystyle \frac{du}{dx}}=2x$.

Now, the derivative of $e^{x^2}$ by the chain rule is equal to

${\displaystyle \frac{d}{dx}}(e^{x^2})$ $={\displaystyle \frac{d}{dx}}(e^u)$

$={\displaystyle \frac{d}{du}}(e^u)\cdot {\displaystyle \frac{du}{dx}}$

$=e^u\cdot 2x$ as du/dx=2x.

$=2x{e}^{x^2}$ as u=x².

Therefore, the differentiation of e to the power x² is equal to the product of 2x and e^{x^2}.

Also Read: 

Derivative of 1 by root(x)

Derivative of root x + 1 by root x

Derivative of x root(x)

Derivative of x sin(x)

Derivative of sin²(x)

Derivative of cos²(x)

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