Advertisements
Advertisements
Question
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
Solution
\[\text{ We have }, f\left( x \right) = \left( 1 + x \right)\left( 1 + x^2 \right)\left( 1 + x^4 \right)\left( 1 + x^8 \right)\]
\[\text {Taking log on both sides }, \]
\[\log f\left( x \right) = \log\left( 1 + x \right) + \log\left( 1 + x^2 \right) + \log\left( 1 + x^4 \right) + \log\left( 1 + x^8 \right)\]
\[ \Rightarrow \frac{d}{dx}\left\{ \log f\left( x \right) \right\} = \frac{d}{dx}\left\{ \log\left( 1 + x \right) + \log\left( 1 + x^2 \right) + \log\left( 1 + x^4 \right) + \log\left( 1 + x^8 \right) \right\}\]
\[ \Rightarrow \frac{1}{f\left( x \right)}f'\left( x \right) = \frac{1}{1 + x} + \frac{2x}{1 + x^2} + \frac{4 x^3}{1 + x^4} + \frac{8 x^7}{1 + x^8}\]
\[ \Rightarrow f'\left( x \right) = \left( 1 + x \right)\left( 1 + x^2 \right)\left( 1 + x^4 \right)\left( 1 + x^8 \right)\left( \frac{1}{1 + x} + \frac{2x}{1 + x^2} + \frac{4 x^3}{1 + x^4} + \frac{8 x^7}{1 + x^8} \right)\]
\[ \Rightarrow f'\left( 1 \right) = \left\{ 1 + \left( 1 \right) \right\}\left\{ 1 + \left( 1 \right)^2 \right\}\left\{ 1 + \left( 1 \right)^4 \right\}\left\{ 1 + \left( 1 \right)^8 \right\}\left\{ \frac{1}{1 + \left( 1 \right)} + \frac{2\left( 1 \right)}{1 + \left( 1 \right)^2} + \frac{4 \left( 1 \right)^3}{1 + \left( 1 \right)^4} + \frac{8 \left( 1 \right)^7}{1 + \left( 1 \right)^8} \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 2 \times 2 \times 2 \times 2\left\{ \frac{1}{2} + \frac{2}{2} + \frac{4}{2} + \frac{8}{2} \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 2 \times 2 \times 2 \times 2 \times \frac{1}{2}\left\{ 1 + 2 + 4 + 8 \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 8 \times 15 = 120\]
APPEARS IN
RELATED QUESTIONS
Differentiate etan x ?
Differentiate `2^(x^3)` ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
Find the second order derivatives of the following function log (sin x) ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If y = etan x, then (cos2 x)y2 =
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
