Advertisements
Advertisements
प्रश्न
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
Advertisements
उत्तर
Here,
\[x = 2\text { at and y } = a t^2 \]
\[\text { Differentiating w . r . t . t, we get }\]
\[\frac{d x}{d t} = 2\text { a and} \frac{d y}{d t} = 2\text { at } \]
\[ \therefore \frac{d y}{d x} = \frac{2at}{2a} = t\]
\[\text { Differentiating again w . r . t . t, we get} \]
\[\frac{d^2 y}{d x^2} = 1 \times \frac{dt}{dx} = \frac{1}{2a}\]
\[\text { Now,} \left[ \frac{d^2 y}{d x^2} \right]_{x = \frac{1}{2}} = \frac{1}{2a}\]
APPEARS IN
संबंधित प्रश्न
Differentiate log7 (2x − 3) ?
Differentiate tan 5x° ?
Differentiate \[3^{e^x}\] ?
Differentiate \[3^{x \log x}\] ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
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 \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\] _________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
Find the second order derivatives of the following function ex sin 5x ?
If \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:
| Year | Jan-March | April-June | July-Sept. | Oct.-Dec. |
| 2010 | 70 | 60 | 45 | 72 |
| 2011 | 79 | 56 | 46 | 84 |
| 2012 | 90 | 64 | 45 | 82 |
Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.
