Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Differentiate the following w.r.t. x:
$(\sin\text{x})^{\cos\text{x}}$
✓

Answer

Let $\text{y}=(\sin\text{x})^{\cos\text{x}}$
$\Rightarrow\ \log\text{y}=\log(\sin\text{x})^{\cos\text{x}}=\cos\text{x}\log(\sin\text{x})$
Differentiate both sides w.r.t.x, we get
$\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{d}}{\text{dx}}=\cos\text{x}\cdot\frac{\text{d}}{\text{dx}}(\log\sin\text{x})+\log\sin\text{x }\frac{\text{d}}{\text{dx}}(\cos\text{x})$
$=\cos\text{x}\cdot\frac{1}{\sin\text{x}}\cdot\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}\cdot(-\sin\text{x})$
$=\cos\text{x}\cdot\cos\text{x}-\log(\sin\text{x})\cdot\sin\text{x}$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\text{y}\big[\cot\text{x}\cos\text{x}-\sin\text{x}\cdot\log(\sin\text{x})\big]$
$=(\sin\text{x})^{\cos\text{x}}\big[\cot\text{x}\cos\text{x}-\sin\text{x}\cdot\log(\sin\text{x})\big]$

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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\cot \text{x}. \text{log}\ \sin\text{x dx}$
→
What inference can you draw if $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$ and $\vec{\text{a}}.\vec{\text{b}}=0.$
→
Find X and Y, if:
  1. $\text {X + Y} = \begin{bmatrix}7&0\\2&5\end{bmatrix} \text {and}\ \text{X} - \text{Y} = \begin{bmatrix}3&0\\0&3\end{bmatrix}$
  2. $2\text {X }+ 3\text {Y} = \begin{bmatrix}2&3\\4&0\end{bmatrix} \text {and}\ 3\text{X }+\text{ 2Y} = \begin{bmatrix}-2&-2\\-1&5\end{bmatrix}$
→
State with reasons whether the following functions have inverse:
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
→
Find the mean and standard deviation of the following probability distributions:
$x_i$ $0$ $1$ $3$ $5$
$p_i$ $0.2$ $0.5$ $0.2$ $0.1$
→
Evaluate the following integrals:
$\int \frac{\text{1}}{\sqrt{\text{x}} + \text{x}} \text{ dx}$
→
Find in vector form as wel as in cartesian form, the equation of the line passing through the points $A(1, 2, -1)$ and $B(2, 1, 1).$
→
Show that the function defined by $g (x) = x – [x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x.$
→
Find the area of the triangle with vertices at the points:
$(0, 0), (6, 0)$ and $(4, 3)$
→
Evaluate the following integrals:
$\frac{\text{x}+1}{\text{x}(1+\text{xe}^\text{x})}\ \text{dx}$
→