Advertisements
Advertisements
प्रश्न
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
विकल्प
1
-1
0
none of these
Advertisements
उत्तर
none of these
\[\text { We have, } f\left( x \right) = \sqrt{x^2 - 10x + 25}\]
\[ = \sqrt{\left( x - 5 \right)^2}\]
\[ = \left| x - 5 \right| \]
`={[x-5 " for " x>5],[-(x-5) " for " x<5]:}`
\[\text { LHD }= \lim_{x \to 5^-} \frac{f\left( x \right) - f\left( a \right)}{x - a}\]
\[ = \lim_{x \to 5^-} \frac{\sqrt{x^2 - 10x + 25} - \sqrt{5^2 - 10\left( 5 \right) + 25}}{x - 5}\]
\[ = \lim_{x \to 5^-} \frac{\left| x - 5 \right|}{x - 5}\]
\[ = \lim_{x \to 5^-} \frac{- \left( x - 5 \right)}{x - 5}\]
\[ = - 1\]
\[RHD = \lim_{x \to 5^+} \frac{f\left( x \right) - f\left( a \right)}{x - a}\]
\[ = \lim_{x \to 5^+} \frac{\sqrt{x^2 - 10x + 25} - \sqrt{5^2 - 10\left( 5 \right) + 25}}{x - 5}\]
\[ = \lim_{x \to 5^+} \frac{\left| x - 5 \right|}{x - 5}\]
\[ = \lim_{x \to 5^+} \frac{x - 5}{x - 5}\]
\[ = 1\]
\[\text { Here, LHD } \neq RHD\]
\[\text { Thus, the function is not differentiable at }x = 5\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate tan 5x° ?
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^{3 x} \cos 2x\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
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 \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[{10}^{ \log \sin x }\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
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 \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?
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) ?
Find the second order derivatives of the following function ex sin 5x ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
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
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
