Advertisements
Advertisements
प्रश्न
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
उत्तर
\[\text{ We have,} f\left( 0 \right) = f\left( 1 \right) = 0 , f'\left( 1 \right) = 2\]
\[\text { and, } \]
\[y = f\left( e^x \right) e^{f\left( x \right)}\]
\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left[ f\left( e^x \right) \times e^{f\left( x \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right)\frac{d}{dx} e^{f\left( x \right)} + e^{f\left( x \right)} \frac{d}{dx}f\left( e^x \right) \left[ \text{Using product rule } \right]\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right) \times e^{f\left( x \right)} \frac{d}{dx}f\left( x \right) + e^{f\left( x \right)} \times f'\left( e^x \right)\frac{d}{dx}\left( e^x \right)\]
\[ \Rightarrow \frac{dy}{dx} = f\left( e^x \right) \times e^{f\left( x \right)} \times f'\left( x \right) + e^{f\left( x \right)} \times f'\left( e^x \right) \times e^x \]
\[\text{ Putting x } = 0, \text{ we get }, \]
\[\frac{dy}{dx} = f\left( e^0 \right) \times e^{f\left( 0 \right)} \times f'\left( 0 \right) + e^{f\left( 0 \right)} \times f'\left( e^0 \right) \times e^0 \]
\[ \Rightarrow \frac{dy}{dx} = f\left( 1 \right) e^{f\left( 0 \right)} \times f'\left( 0 \right) + e^{f\left( 0 \right)} \times f'\left( 1 \right) \times 1\]
\[ \Rightarrow \frac{dy}{dx} = 0 \times e^0 \times f'\left( 0 \right) + e^0 \times 2 \times 1 .........\left[ \because f\left( x \right) = f\left( 1 \right) = 0 \text{ and }f'\left( 1 \right) = 2 \right]\]
\[ \Rightarrow \frac{dy}{dx} = 0 + 1 \times 2 \times 1\]
\[ \Rightarrow \frac{dy}{dx} = 2\]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles log cosec x ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
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 \[e^{x \log x}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
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 \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
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 \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
Differentiate sin(log sin x) ?
If p, q, r, s are real number and pr = 2(q + s) then for the equation x2 + px + q = 0 and x2 + rx + s = 0 which of the following statement is true?
