Advertisements
Advertisements
प्रश्न
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Advertisements
उत्तर
\[\text{ Let } f\left( x \right) = e^\sqrt{2x} \]
\[ \Rightarrow f\left( x + h \right) = e^\sqrt{2\left( x + h \right)} \]
\[ \therefore \frac{d}{dx}\left\{ f\left( x \right) \right\} = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{e^{{}^\sqrt{2\left( x + h \right)}} - e^{{}^\sqrt{2x}}}{h}\]
\[ = \lim_{h \to 0} e^\sqrt{2x} \left[ \frac{e^\sqrt{2\left( x + h \right)} - \sqrt{2x} - 1}{h} \right]\]
\[ = e^\sqrt{2x} \lim_{h \to 0} \left[ \frac{e^\sqrt{2\left( x + h \right)} - \sqrt{2x} - 1}{\sqrt{2\left( x + h \right)} - \sqrt{2x}} \right] \times \lim_{h \to 0} \frac{\sqrt{2\left( x + h \right)} - \sqrt{2x}}{h}\]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{\sqrt{2\left( x + h \right)} - \sqrt{2x}}{h} \left[ \because \lim_{h \to 0} \frac{e^h - 1}{h} = 1 \right]\]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{\sqrt{2\left( x + h \right)} - \sqrt{2x}}{h} \times \frac{\sqrt{2\left( x + h \right)} + \sqrt{2x}}{\sqrt{2\left( x + h \right)} + \sqrt{2x}} \] [Rationalising the numerator]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{2\left( x + h \right) - 2x}{h\left( \sqrt{2\left( x + h \right)} + \sqrt{2x} \right)}\]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{2x + 2h - 2x}{h\left( \sqrt{2\left( x + h \right)} + \sqrt{2x} \right)} \]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{2h}{h\left( \sqrt{2\left( x + h \right)} + \sqrt{2x} \right)}\]
\[ = e^\sqrt{2x} \lim_{h \to 0} \frac{2}{\left( \sqrt{2\left( x + h \right)} + \sqrt{2x} \right)}\]
\[ = \frac{e^\sqrt{2x}}{\sqrt{2x}}\]
\[\text{ Hence }, \frac{d}{dx}\left( e^\sqrt{2x} \right) = \frac{e^\sqrt{2x}}{\sqrt{2x}}\]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate the following functions from first principles e−x.
Differentiate \[3^{x \log x}\] ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
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 \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
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 \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
If \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
Differential coefficient of sec(tan−1 x) is ______.
If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 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=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
