Name: Derivative of Functions in Parametric Form
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: Learn everything you'll need to know about derivatives of functions in parametric forms with this quick and easy lesson that'll cover the concept comprehensi...

Question

Differentiate

\sin^{- 1} (4 x \sqrt{1 - 4 x^{2}})

with respect to

\sqrt{1 - 4 x^{2}}

, if

x \in (- \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}})

?

Accepted Answer

Let, $u = \sin^{- 1} (4 x \sqrt{1 - 4 x^{2}})$

put $2 x = \cos θ$

$\Rightarrow u = \sin^{- 1} (2 \times \cos θ \sqrt{1 - \cos^{2} θ})$

$\Rightarrow u = \sin^{- 1} (2 \cos θ \sin θ)$

$\Rightarrow u = \sin^{- 1} (\sin 2 θ)$ .........(i)

Let, $v = \sqrt{1 - 4 x^{2}}$ .........(ii)

Here,

$x \in (- \frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}})$

$\Rightarrow 2 x \in (- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$

$\Rightarrow θ \in (\frac{π}{4}, \frac{3 π}{4})$

So, from equation (i),

$u = π - 2 θ . . . . . [Since, \sin^{- 1} (\sin θ) = π - θ, if θ \in (\frac{π}{2}, π)]$

$\Rightarrow u = π - 2 \cos^{- 1} (2 x) [Since, 2 x = \cos θ]$

Differentiating it with respect to $x$ ,

$\frac{d u}{d x} = 0 - 2 (\frac{- 1}{\sqrt{1 - {(2 x)}^{2}}}) \frac{d}{d x} (2 x)$

$\Rightarrow \frac{d u}{d x} = \frac{2}{\sqrt{1 - 4 x^{2}}} (2)$

$\Rightarrow \frac{d u}{d x} = \frac{4}{\sqrt{1 - 4 x^{2}}}$ (iii)

from equation (ii)

$\frac{d v}{d x} = \frac{- 4 x}{\sqrt{1 - 4 x^{2}}}$

but, $x \in (- \frac{1}{2}, - \frac{1}{2 \sqrt{2}})$

$\frac{d v}{d x} = \frac{- 4 (- x)}{\sqrt{1 - 4 {(- x)}^{2}}}$

$\Rightarrow \frac{d v}{d x} = \frac{4 x}{\sqrt{1 - 4 x^{2}}}$ (iv)

Diferentiating equation (ii) with respect to $x$ ,

$\frac{d v}{d x} = \frac{1}{2 \sqrt{1 - 4 x^{2}}} \frac{d}{d x} (1 - 4 x^{2})$

$\Rightarrow \frac{d v}{d x} = \frac{1}{2 \sqrt{1 - 4 x^{2}}} (- 8 x)$

$\Rightarrow \frac{d v}{d x} = \frac{- 4 x}{\sqrt{1 - 4 x^{2}}}$ .......(v)

Dividing equation (iii) by (v)

$\frac{\frac{d u}{d x}}{\frac{d v}{d x}} = \frac{4}{\sqrt{1 - 4 x^{2}}} \times \frac{\sqrt{1 - 4 x^{2}}}{- 4 x}$

$∴ \frac{d u}{d v} = - \frac{1}{x}$

Differentiate $\sin^{- 1} (4 x \sqrt{1 - 4 x^{2}})$ with respect to $\sqrt{1 - 4 x^{2}}$ , if $x \in (- \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}})$ ?

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate sin-1 4x1-4x2 with respect to 1-4x2 , if x∈-122,122 ?

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate $\sin^{- 1} (4 x \sqrt{1 - 4 x^{2}})$ with respect to $\sqrt{1 - 4 x^{2}}$ , if $x \in (- \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}})$ ?