Differentiate x^ x x + x^ 2 +1 x^ 2 -1 w.r.t.x. - Vidyadip

Question

Differentiate $x^{x \cos x}$ + $\frac{\text{x}^{2}+1}{\text{x}^{2}-1}$ w.r.t.x.

✓

Answer

$\text{y}=\text{x}^{\text{x cos x}}+\frac{\text{x}^{2}+1}{\text{x}^{2}-1}=\text{u + v}$$\text{u}=\text{x}^{\text{x cos x}}\Rightarrow\log\text{u}=\text{x cos x}\cdot{\text{log x}}$
$\therefore\frac{1}{\text{u}}\cdot\frac{\text{du}}{\text{dx}}=\frac{\text{x cos x}}{\text{x}}+\text{cos x}\cdot{\text{log x - sin x log x}}$
$\Rightarrow$$\frac{\text{du}}{\text{dx}}=\text{x}^{\text{x cos x}}(\text{cos x+cos x log x}\text{ - sin x log x})$
$\text{v}=\frac{\text{x}^{2}+1}{\text{x}^{2}-1},\frac{\text{dv}}{\text{dx}}=\frac{(\text{x}^{2}-1)\text{2x}-\text{(x}^{2}+1)\text{2x}}{\text{(x}^{2}-1)^{2}}=-\frac{\text{4x}}{\text{(x}^{2}-1)^{2}}$
$\therefore\frac{\text{dy}}{\text{dx}}=\text{x}^{\text{x cos x}}\text{(cos x + log x}\cdot\text{cos x - sin x log x})-\frac{\text{4x}}{(\text{x}^{2}-1)^{2}}$

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

Verify the Rolle’s theorem for each of the functions:
$f(x) = x(x - 1)^2$ in $[0, 1].$

Solve the following initial value problems:
$(1+\text{y}^2)\text{dx}+(\text{x}-\text{e}^{\tan^{-1}\text{y}})\text{dy}=0,\text{ y}(0)=0$

Maximum Z = 15x + 10y
Subject to
$3\text{x}+2\text{y}\leq80$
$2\text{x}+3\text{y}\leq70$
$\text{x},\text{y}\geq0$

If $\text{A}=\begin{bmatrix}1&0&-2\\3&-1&0\\-2&1&1\end{bmatrix},\text{B}=\begin{bmatrix}0&5&-4\\-2&1&3\\-1&0&2\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&5&2\\-1&1&0\\0&-1&1\end{bmatrix},$ verify that A(B - C) = AB - AC.

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

Using integration, find the area of the region: $\left\{(\text{x},\text{y}) : |\text{x}-1|<\text{y}<\sqrt{5-\text{x}^{2}}\right\}$.

If the marginal cost of maufacturing a certain item is given by $\text{C}(\text{x})=\frac{\text{dC}}{\text{dx}}=2+0.15\text{x}$. Find the total cost function C(x), given that C(0) = 100.

Using elementary operations, find the inverse of the following matrix:$\begin{pmatrix} -1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{pmatrix}$

show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$