Advertisements
Advertisements
प्रश्न
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)` ?
Advertisements
उत्तर
\[\text{ We have }, f\left( x \right) = \left( 1 + x \right)\left( 1 + x^2 \right)\left( 1 + x^4 \right)\left( 1 + x^8 \right)\]
\[\text {Taking log on both sides }, \]
\[\log f\left( x \right) = \log\left( 1 + x \right) + \log\left( 1 + x^2 \right) + \log\left( 1 + x^4 \right) + \log\left( 1 + x^8 \right)\]
\[ \Rightarrow \frac{d}{dx}\left\{ \log f\left( x \right) \right\} = \frac{d}{dx}\left\{ \log\left( 1 + x \right) + \log\left( 1 + x^2 \right) + \log\left( 1 + x^4 \right) + \log\left( 1 + x^8 \right) \right\}\]
\[ \Rightarrow \frac{1}{f\left( x \right)}f'\left( x \right) = \frac{1}{1 + x} + \frac{2x}{1 + x^2} + \frac{4 x^3}{1 + x^4} + \frac{8 x^7}{1 + x^8}\]
\[ \Rightarrow f'\left( x \right) = \left( 1 + x \right)\left( 1 + x^2 \right)\left( 1 + x^4 \right)\left( 1 + x^8 \right)\left( \frac{1}{1 + x} + \frac{2x}{1 + x^2} + \frac{4 x^3}{1 + x^4} + \frac{8 x^7}{1 + x^8} \right)\]
\[ \Rightarrow f'\left( 1 \right) = \left\{ 1 + \left( 1 \right) \right\}\left\{ 1 + \left( 1 \right)^2 \right\}\left\{ 1 + \left( 1 \right)^4 \right\}\left\{ 1 + \left( 1 \right)^8 \right\}\left\{ \frac{1}{1 + \left( 1 \right)} + \frac{2\left( 1 \right)}{1 + \left( 1 \right)^2} + \frac{4 \left( 1 \right)^3}{1 + \left( 1 \right)^4} + \frac{8 \left( 1 \right)^7}{1 + \left( 1 \right)^8} \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 2 \times 2 \times 2 \times 2\left\{ \frac{1}{2} + \frac{2}{2} + \frac{4}{2} + \frac{8}{2} \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 2 \times 2 \times 2 \times 2 \times \frac{1}{2}\left\{ 1 + 2 + 4 + 8 \right\}\]
\[ \Rightarrow f'\left( 1 \right) = 8 \times 15 = 120\]
APPEARS IN
संबंधित प्रश्न
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate etan x ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Differentiate x2 with respect to x3
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to _____________ .
Find the second order derivatives of the following function sin (log x) ?
Find the second order derivatives of the following function x3 log x ?
Find the second order derivatives of the following function tan−1 x ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
