Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
Differentiate the functions given in Exercise:
$\text{x}^\text{x}-2^{\sin\text{x}}$
✓

Answer

Let $\text{y}=\text{x}^\text{x}-2^{\sin\text{x}}$
Putting $\text{u}=\text{x}^\text{x}\text{ and v }=2^{\sin\text{x}}$
$\Rightarrow\ \text{y}=\text{u}-\text{v}\ \Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{du}}{\text{dx}}-\frac{\text{dv}}{\text{dx}}\ \dots\text{(i)}$
Now, u = xx $\ \Rightarrow\ \log\text{u}=\log\text{x}^\text{x}=\text{x}\log\text{x}$
$\therefore\ \frac{\text{d}}{\text{dx}}\log\text{u}=\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})$ $\Rightarrow\ \frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}\log\text{x}+\log\text{x}\frac{\text{d}}{\text{dx}}\text{x}$
$\Rightarrow\ \frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=\text{x}\frac{1}{\text{x}}+\log\text{x}.1$ $\Rightarrow\ \frac{1}{\text{u}}\frac{\text{du}}{\text{dx}}=1+\log\text{x}$
$\Rightarrow\ \frac{\text{du}}{\text{dx}}=\text{u}(1+\log\text{x})$ $=\text{x}^\text{x}(1+\log\text{x})\ \dots\text{(ii)}$
Again, $\text{v}=2^{\sin\text{x}}\ \Rightarrow\ \frac{\text{dv}}{\text{dx}}=\frac{\text{d}}{\text{dx}}2^{\sin\text{x}}$
$\Rightarrow\ \frac{\text{dv}}{\text{dx}}=2^{\sin \text{x}}\log2\frac{\text{d}}{\text{dx}}\sin\text{x}=\ \Big[\because\frac{\text{d}}{\text{dx}}\text{a}^{\text{f(x)}}=\text{a}^{\text{f(x)}}\log\text{a}\frac{\text{d}}{\text{dx}}\text{f(x)}\Big]$
$\frac{\text{dv}}{\text{dx}}=2^{\sin\text{x}}(\log2).\cos\text{x}=\cos\text{x}.2^{\sin\text{x}}\log2\ \dots\text{(iii)}$
Putting the values from eq. (ii) and (iii) in eq. (i),
$\frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}(1+\log\text{x})-\cos\text{x}.2^{\sin\text{x}}\log2$

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

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically.
→
Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.
→
If R and S are relations on a set A, then prove that:
  1. R and S are symmetric $\Rightarrow\ \text{R}\cap\text{S}$ and $\text{R}\cup\text{S}$ are symmetric
  2. R is reflexive and S is any relation $\Rightarrow\ \text{R}\cup\text{S}$ is reflexive.
→
Express the following matrix as the sum of a symmetric and a skew symmetric matrix, and verify your result:
$\begin{pmatrix}3&-2&-4\\3&-2&-5\\-1&1&2\end{pmatrix}$
→
Verify Rolle's theorem for the following function on the indicated intervals

f(x) = x2 + 5 x + 6 on the interval [-3, -2]

→
Calculate the area of the ragion bounded by y2 = x and x2 = y.
→
Find the equation of the perpendicular drawn from the point P(-1, 3, 2) to the line $\vec{\text{r}}=\big(2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the coordinates of the foot of the perpendicular from P.
→
Find the maximum and the minimum values, if any, without using derivaives of the following functions:

f(x) = x3 - 1 on R.

→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(3\hat{\text{i}}+5\hat{\text{j}}+7\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}\big)$ and $\vec{\text{r}}=-\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(7\hat{\text{i}}-6\hat{\text{j}}+\hat{\text{k}}\big)$
→
If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that A-1 = A3.
→