Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Differentiate the following functions with respect to x:
$\sin(\text{x}^\text{x})$
✓

Answer

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

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 "5 Marks Questions" 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

A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$
→
Show that $\text{y}=\text{A}\cos2\text{x}+\text{B}\sin2\text{x}$ is a solution of the differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$
→
$\text{Find}:\int \frac{\sin \theta\ \text{d}\theta}{(4 + \cos^{2} \theta) (2- \sin^{2} \theta)} \text{d} \theta$
→
If a relation R is defined on the set Z of integers as follows: $(a, b) \in R ⇔ a^2 + b^2 = 25$. Then, domain (R) is:
→
If a, b, c are the langths of sides, BC, CA and AB of a triangle ABC, prove that $\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}+\overrightarrow{\text{AB}}=\vec{\text{0}}$ and deduce that $\frac{\text{a}}{\sin\text{A}}=\frac{\text{b}}{\sin\text{B}}=\frac{\text{c}}{\sin\text{C}}.$
→
A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240kg of phosphoric acid, at least 270kg of potash and at most 310kg of chlorine.
Kg per bag
 
Brand P
Brand P
Nitrogen
32
3.5
Phosphoric
1
2
Potash
3
1.5
Chlorine
1.5
2
If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?
→
Find the curve for which the intercept cut-off by a tangent on x-axis is equal to four times the ordinate of the point of contact.
→
For the matrices A and B, verify that (AB)' = B'A' where
  1. $\text{A}=\begin{bmatrix}1\\-4\\3\end{bmatrix},\text{B}=\begin{bmatrix}-1&2&1\end{bmatrix}$
  2. $\text{A}=\begin{bmatrix}0\\1\\2\end{bmatrix},\text{B}=\begin{bmatrix}1&5&7\end{bmatrix}$
→
A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?
→