Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{ We have, y }= x^n + n^x + x^x + n^n \]
\[ \Rightarrow y = x^n + n^x + e^{\log x^x }+ n^n \]
\[ \Rightarrow y = x^n + n^x + e^{x\log x} + n^n\]
Differentiate with respect to x,
\[\frac{dy}{dx} = \frac{d}{dx}\left( x^n \right) + \frac{d}{dx}\left( n^x \right) + \frac{d}{dx}\left( e^{x\log x} \right) + \frac{d}{dx}\left( n^n \right)\]
\[ = n x^{n - 1} + n^x \log n + e^{\log x^x} \left[ x\frac{d}{dx}\log x + \log x\frac{d}{dx}\left( x \right) \right]\]
\[ = n x^{n - 1} + n^x \log n + x^x \left[ x\left( \frac{1}{x} \right) + \log x \right]\]
\[ = n x^{n - 1} + n^x \log n + x^x \left[ 1 + \log x \right]\]
\[ = n x^{n - 1} + n^x \log n + x^x \left[ \log e + \log x \right] .........\left[ \because \log_e e = 1 \text{ and }\log A + \log B = \log\left( AB \right) \right]\]
\[ = n x^{n - 1} + n^x \log n + x^x \log\left( ex \right)\]
APPEARS IN
संबंधित प्रश्न
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Differentiate the following functions from first principles e3x.
Differentiate sin (log x) ?
Differentiate logx 3 ?
Differentiate (log sin x)2 ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
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 \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left(36 \right)^x} \right\}\] with respect to x.
If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
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 \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
Find the second order derivatives of the following function ex sin 5x ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
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 y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]
Disclaimer: There is a misprint in the question,
\[\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0\] must be written instead of
\[\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
