Name: Differentiation of Exponential Functions
Uploaded: 2019-07-19
Duration: 12 min 3 s
Description: To know how to find the derivative of exponential functions or multiple functions containing the exponential term. Let's watch this video and explore!

Question

Differentiate the following w.r.t. $x :$

$\frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log_{e} (\sqrt{1 - x^{2}}), |x| < 1$

Accepted Answer

Let $y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log_{e} (\sqrt{1 - x^{2}}), |x| < 1$

Differentiate the above equation with respect to $x,$ we get

$\Rightarrow \frac{d y}{d x} = \frac{d}{d x} [\frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log_{e} (\sqrt{1 - x^{2}})]$

$\Rightarrow \frac{d y}{d x} = [\frac{d}{d x} \{\frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}}\} + \frac{d}{d x} \{\log_{e} (\sqrt{1 - x^{2}})\}]$

As we know, by Quotient rule $\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{g (x) f^{'} (x) - f (x) g^{'} (x)}{g^{2} (x)}$ and by Chain rule $\frac{d}{d x} (h (k (x))) = h^{'} (k (x)) \frac{d}{d x} (k (x))$

So, $\frac{d y}{d x} = [\{\frac{\sqrt{1 - x^{2}} \frac{d}{d x} (x \sin^{- 1} x) - (x \sin^{- 1} x) \frac{d}{d x} (\sqrt{1 - x^{2}})}{{(\sqrt{1 - x^{2}})}^{2}}\} + \frac{1}{\sqrt{1 - x^{2}}} \{\frac{d}{d x} (\sqrt{1 - x^{2}})\}]$

$\Rightarrow \frac{d y}{d x} = [\{\frac{\sqrt{1 - x^{2}} (x \frac{d}{d x} (\sin^{- 1} x) + \sin^{- 1} x \frac{d}{d x} (x)) - (x \sin^{- 1} x) (\frac{1}{2 \sqrt{1 - x^{2}}} \frac{d}{d x} (1 - x^{2}))}{1 - x^{2}}\} + \frac{1}{\sqrt{1 - x^{2}}} \{\frac{1}{2 \sqrt{1 - x^{2}}} \frac{d}{d x} (1 - x^{2})\}]$

$\{∵ \frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + g^{'} (x) f (x)\}$

$\Rightarrow \frac{d y}{d x} = [\{\frac{\sqrt{1 - x^{2}} (\frac{x}{\sqrt{1 - x^{2}}} + \sin^{- 1} x) - (x \sin^{- 1} x) (\frac{- 2 x}{2 \sqrt{1 - x^{2}}})}{1 - x^{2}}\} + \frac{1}{\sqrt{1 - x^{2}}} \{\frac{- 2 x}{2 \sqrt{1 - x^{2}}}\}]$

$\{∵ \frac{d}{d x} (\sin^{- 1} x) = \frac{1}{\sqrt{1 - x^{2}}}, \frac{d}{d x} (x^{n}) = n x^{n - 1}\}$

$\Rightarrow \frac{d y}{d x} = [\{\frac{(x + \sqrt{1 - x^{2}} \sin^{- 1} x) + (\frac{x^{2} \sin^{- 1} x}{\sqrt{1 - x^{2}}})}{1 - x^{2}}\} - \{\frac{x}{1 - x^{2}}\}]$

$\Rightarrow \frac{d y}{d x} = [\{\frac{(x \sqrt{1 - x^{2}} + (1 - x^{2}) \sin^{- 1} x) + (x^{2} \sin^{- 1} x)}{(1 - x^{2}) (\sqrt{1 - x^{2}})}\} - \{\frac{x}{1 - x^{2}}\}]$

$\Rightarrow \frac{d y}{d x} = \frac{\sin^{- 1} x}{{(1 - x^{2})}^{\frac{3}{2}}}$ .

Differentiate the following w.r.t. $x :$

$\frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log_{e} (\sqrt{1 - x^{2}}), |x| < 1$

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following w.r.t. x: xsin-1x1-x2+loge1-x2, x<1

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following w.r.t. $x :$

$\frac{x \sin^{- 1} x}{\sqrt{1 - x^{2}}} + \log_{e} (\sqrt{1 - x^{2}}), |x| < 1$