Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$(\text{x}^\text{x})\sqrt{\text{x}}$
✓

Answer

Let $\text{y}=(\text{x}^\text{x})\sqrt{\text{x}}\ .....(\text{i})$
Taking log on both the sides,
$\log\text{y}=\log(\text{x}^\text{x}\sqrt{\text{x}})$
$\log\text{y}=\text{x}\log\text{x}+\frac{1}{2}\log\text{x}$
Differentiating it with respect to x,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})+\frac{1}{2}\frac{\text{d}}{\text{dx}}(\log\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{x}\Big(\frac{1}{\text{x}}\Big)+\log\text{x}(1)+\frac{1}{2}\Big(\frac{1}{\text{x}}\Big)$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=1+\log\text{x}+\frac{1}{2\text{x}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big[1+\log\text{x}+\frac{1}{2\text{x}}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}\sqrt{\text{x}}\Big[1+\log\text{x}+\frac{1}{2\text{x}}\Big]$
[Using equation (i)]
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}^{\text{x}+\frac{1}{2}}\Big[\Big(\frac{2\text{x}+1}{2\text{x}}\Big)+\log\text{x}\Big]$

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 "Solve the Following Question.(5 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Discuss the continuity and differentiability of,$\text{f(x)}=\begin{cases}(\text{x}-\text{c})\cos\Big(\frac{1}{\text{x}-\text{c}}\Big), & \text{x}\neq 0\\0, & \text{x}= 0\end{cases}$
→
A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
A = the card drawn is a spade,
B = the card drawn in an ace.
→
Using differentials, find the approximate values of the following:
$\sqrt{26}$
→
If $\text{y}=(\sin\text{x}-\cos\text{x})^{\sin\text{x}-\cos\text{x}},\frac{\pi}{4}<\text{x}<\frac{3\pi}{4},$ find $\frac{\text{dy}}{\text{dx}}$
→
Let $\bar{b}=4 \hat{i}+3 \hat{j}$ and $\bar{c}$ be two vectors perpendicular to each other in the $X Y$-plane. Find

vectors in the same plane having projection 1 and 2 along $\bar{b}$ and $\bar{c}_{\text {, respectively, are given Y }}$

→
Show that:
$\begin{vmatrix}\text{y}+\text{z}&\text{x}&\text{y}\\\text{z}+\text{x}&\text{z}&\text{x}\\\text{x}+\text{y}&\text{y}&\text{z}\end{vmatrix}=(\text{x}+\text{y}+\text{z})(\text{x}-\text{z})^3$
→
Show that the following systems of linear equations has infinite number of solutions and solve:
$2x + y - 2z = 0,$
$x - 2y + z = -2,$
$5x - 5y + z = -2$
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{3}{2}}\text{x}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}\text{ dx}$
→
Find the area bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4:$
By using vertical strips.
→
Show that the following system of linear equations is consistent and also find solution:
$x + y + z = 6$
$x + 2y + 3z = 14$
$x + 4y + 7z = 30$
→