Advertisements
Advertisements
Question
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
Advertisements
Solution
\[\text{ We have,} f\left( 0 \right) = f\left( 1 \right) = 0 , f'\left( 1 \right) = 2\]
\[\text { and, } \]
\[y = f\left( e^x \right) e^{f\left( x \right)}\]
\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left[ f\left( e^x \right) \times e^{f\left( x \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right)\frac{d}{dx} e^{f\left( x \right)} + e^{f\left( x \right)} \frac{d}{dx}f\left( e^x \right) \left[ \text{Using product rule } \right]\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right) \times e^{f\left( x \right)} \frac{d}{dx}f\left( x \right) + e^{f\left( x \right)} \times f'\left( e^x \right)\frac{d}{dx}\left( e^x \right)\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right) \times e^{f\left( x \right)} \times f'\left( x \right) + e^{f\left( x \right)} \times f'\left( e^x \right) \times e^x \]
\[\text{ Putting x } = 0, \text{ we get }, \]
\[\frac{dy}{dx} = f\left( e^0 \right) \times e^{f\left( 0 \right)} \times f'\left( 0 \right) + e^{f\left( 0 \right)} \times f'\left( e^0 \right) \times e^0 \]
\[ \Rightarrow \frac{dy}{dx} = f\left( 1 \right) e^{f\left( 0 \right)} \times f'\left( 0 \right) + e^{f\left( 0 \right)} \times f'\left( 1 \right) \times 1\]
\[ \Rightarrow \frac{dy}{dx} = 0 \times e^0 \times f'\left( 0 \right) + e^0 \times 2 \times 1 .........\left[ \because f\left( x \right) = f\left( 1 \right) = 0 \text{ and }f'\left( 1 \right) = 2 \right]\]
\[ \Rightarrow \frac{dy}{dx} = 0 + 1 \times 2 \times 1\]
\[ \Rightarrow \frac{dy}{dx} = 2\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles e3x.
Differentiate the following functions from first principles log cos x ?
Differentiate the following functions from first principles log cosec x ?
Differentiate sin (log x) ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left(36 \right)^x} \right\}\] with respect to x.
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 \[{10}^\left( {10}^x \right)\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
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)` ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
Differential coefficient of sec(tan−1 x) is ______.
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
Find the second order derivatives of the following function sin (log x) ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of ar, 0 < r ≤ n, is equal to
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
