Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$10^{(10 x)}$

Solution:

Let $y=10^{10 x}$

Taking log both the sides:

$\Rightarrow \log y=\log 10^{10 x}$

$\Rightarrow \log y=10 x \log 10\left\{\log x^{a}=\operatorname{alog} x\right\}$

$\Rightarrow \log y=(10 \log 10) x$

Differentiating with respect to $x$ :

$\Rightarrow \frac{d(\log y)}{d x}=\frac{d\{(10 \log 10) x\}}{d x}$

$\Rightarrow \frac{\mathrm{d}(\log \mathrm{y})}{\mathrm{dx}}=10 \times \log (10) \times \frac{\mathrm{d}(\mathrm{x})}{\mathrm{dx}}\{$ Here $10 \log (10)$ is a constant term $\}$

$\left\{\right.$ Using chain rule, $\frac{\mathrm{d}(\mathrm{au})}{\mathrm{dx}}=\mathrm{a} \frac{\mathrm{du}}{\mathrm{dx}}$ where $\mathrm{a}$ is any constant and $\mathrm{u}$ is any variable $\}$

$\Rightarrow \frac{1}{y} \frac{d y}{d x}=10 \log (10)$

$\left\{\frac{d(\log u)}{d x}=\frac{1}{u} \frac{d u}{d x} ; \frac{d(\sin x)}{d x}=\cos x\right\}$

$\Rightarrow \frac{1}{y} \frac{d y}{d x}=10 \log (10)$

$\Rightarrow \frac{d y}{d x}=y\{10 \log (10)\}$

Put the value of $y=10^{10} x$ :

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=10^{10 \mathrm{x}}\{10 \log (10)\}$