Question

Differentiate the following functions with respect to $x$ :

$\frac{{10}^x}{\sin x}$

Answer 1

Let, $y=\frac{{10}^x}{\sin x}$

$\Rightarrow \frac{dy}{dx}=\frac{d}{dx}\left(\frac{{10}^x}{sinx}\right)$

Quotient rule: $\frac{d}{dx}\left(\frac{U}{V}\right)=\frac{V\frac{dU}{dx}-U\frac{dV}{dx}}{V^2}$

$\Rightarrow \frac{dy}{dx}=\frac{\left(\sin x\right)\frac{d}{dx}\left({10}^x\right)-\left({10}^x\right)\frac{d}{dx}\left(\sin x\right)}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{\sin x\times {10}^x\log 10-{10}^x\cos x}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{\sin x\times {10}^x\log 10}{(\sin x{)}^2}-\frac{{10}^x\cos x}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{{10}^x\log 10}{\sin x}-\frac{{10}^x\cos x}{\sin x\times \sin x}$

$\Rightarrow \frac{dy}{dx}={10}^x\text{cosec~x}\log 10-{10}^x\text{cosec}x\cot x$

$\Rightarrow \frac{dy}{dx}={10}^x\text{cosec}x(\log 10-\cot x)$

$\therefore \frac{d}{dx}\left(\frac{{10}^x}{\sin x}\right)={10}^x\text{cosec}x(\log 10-\cot x)$