Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Differentiate the following function with respect to x:

$(\log\text{x})^{x} + \text{x}^{\log\text{x}}.$

✓

Answer

Let y =(log x)x + x logx
$\Rightarrow$ y =u+v where u=(log x)x , v = xlog x
$\Rightarrow\frac{\text{dy}}{\text{dx}} = \frac{\text{du}}{\text{dx}} + \frac{\text{dv}}{\text{dx}}$ - - - -  (i)
Now u=(log x)x
Taking logarithm of both sides, we get
logu= x. log(log x)
Differentiating both sides w.r.t.x, we get
$\frac{1}{\text{u}}.\frac{\text{du}}{\text{dx}} =\text{x}.\frac{1}{\log\text{x}}.\frac{1}{\text{x}} + \log(\log\text{x} )\Rightarrow\frac{\text{du}}{\text{dx}} = \text{u}\left\{\frac{1}{\log\text{x}} + \log(\log\text{x})\right\}$
$\Rightarrow\frac{\text{du}}{\text{dx}} = (\log\text{x})^{x}\left\{\frac{1}{\log\text{x}} + \log(\log\text{x})\right\}$ -  - - -  (ii)
Again v = xlog x
Taking logarithm of both sides , we get
log v = log xlog x
$\Rightarrow\log\text{v} = \log\text{x}.\log\text{x}\Rightarrow\log v = ( \log\text{x})^{2}$
Differentiating both sides w.r.t.x, we get
$\frac{1}{v}\frac{\text{dv}}{\text{dx}} = 2 \log\text{x}.\frac{1}{\text{x}}$
$\Rightarrow\frac{\text{dv}}{\text{dx}} = 2 \text{x}^{log\text{ x}}.\frac{\log\text{x}}{\text{x}}$ - - - - - (iii)
Putting $\frac{\text{du}}{\text{dx}}\text{ and } \frac{\text{dv}}{\text{dx}}$ from (ii) and (iii) in (i) we get
$\frac{\text{dy}}{\text{dx}} = ( \log\text{x})^{x}\left\{\frac{1}{\log\text{x}} + \log(\log\text{x})\right\} + 2 \frac{\log\text{x.x}^{\log\text{x}}}{\text{x}}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluvate the following intregals
$\int\frac{2\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+5}}\text{dx}$
→
Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4.
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}(2^{\text{x}}+2^{-\text{x}})^2&(2^{\text{x}}-2^{-\text{x}})^2&1\$3^{\text{x}}+3^{-\text{x}})^2&(3^{\text{x}}-3^{-\text{x}})^2&1\$4^{\text{x}}+4^{-\text{x}})^2&(4^{\text{x}}-4^{-\text{x}})^2&1\end{vmatrix}$
→
Maximize Z = 9x + 3y
Subject to
$2\text{x}+3\text{y}\leq13$
$3\text{x}+\text{y}\leq5$
$\text{x},\text{y}\geq0$
→
Evaluate the following integrals:
$\int(\text{e}^{\log\text{x}}+\sin\text{x})\cos\text{x dx}$
→
Solve the following equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$
→
Evaluate the following integrals:
$\int\limits^{(\pi)^\frac{2}{3}}_{0}\sqrt{\text{x}}\cos^2\text{x}^{\frac{3}{2}}\text{ dx}$
→
$\text{If y = 3} \cos (\log\text{x}) + 4\sin (\log \text{x}), \text{then show that x}^{2} .\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}} + \text{y} = 0$
→
$\int\frac{\text{x}-1}{\sqrt{\text{x}+4}}\ \text{dx}$
→
Find the general solution of (1 + tany) (dx - dy) + 2xdy = 0.
→