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 ____________ .
पर्याय
1
-1
0
none of these
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 the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate the following functions from first principles log cosec x ?
Differentiate sin (log x) ?
Differentiate tan 5x° ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\cos \left( \log x \right)^2\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[{10}^{ \log \sin x }\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
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)` ?
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
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 \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
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 \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
Find the second order derivatives of the following function x3 + tan x ?
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
If y = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
If xy − loge y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]
