Find dy dx y=x^ x +( )^x - Vidyadip

Question

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\text{x}}+(\sin\text{x})^\text{x}$

✓

Answer

We have, $\text{y}=\text{x}^{\text{x}}+(\sin\text{x})^\text{x}$
$\Rightarrow\text{y}=\text{e}^{\log\text{x}^\text{x}}+\text{e}^{\log(\sin\text{x})^\text{x}}$
$\Rightarrow\text{y}=\text{e}^{\text{x}\log\text{x}}+\text{e}^{\text{x}\log\sin\text{x}}$
Differentiating with respect to x using chain rule and product rule,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\text{x}\log\text{x}}\big)+\frac{\text{d}}{\text{dx}}\big(\text{e}^{\text{x}\log\sin\text{x}}\big)$
$=\text{e}^{\text{x}\log\text{x}}\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})+\text{e}^{\text{x}\log\sin\text{x}}\frac{\text{d}}{\text{dx}}(\text{x}\log\sin\text{x})$
$=\text{e}^{\text{x}\log\text{x}}\Big[\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big] \\ +\text{e}^{\log(\sin\text{x})^\text{x}}\Big[\text{x}\frac{\text{d}}{\text{dx}}(\log\sin\text{x})+\log\sin\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big]$
$=\text{x}^{\text{x}}\Big[\text{x}\Big(\frac{1}{\text{x}}\Big)+\log\text{x}(1)\Big] \\ +(\sin\text{x})^\text{x}\Big[\text{x}\Big(\frac{1}{\sin\text{x}}\Big)\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}\big]$
$=\text{x}^{\text{x}}\big[1+\log\text{x}\big]+(\sin\text{x})^\text{x}\Big[\text{x}\Big(\frac{1}{\sin\text{x}}\Big)(\cos\text{x})+\log\sin\text{x}\Big]$
$=\text{x}^{\text{x}}\big[1+\log\text{x}\big]+(\sin\text{x})^\text{x}\big[\text{x}\cot\text{x}+\log\sin\text{x}\big]$

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 "5 Marks Questions" from Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

verify that $\text{y}=\log(\text{x}+\sqrt{\text{x}^2+\text{a}^2})^2$ is a solution of the differential equation $(\text{a}^2+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}=0$

Differentiate $(\log\text{x})^\text{x}$ with respect to x.

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&\text{a}&\text{a}^2-\text{bc}\\1&\text{b}&\text{b}^2-\text{ac}\\1&\text{c}&\text{c}^2-\text{ab} \end{vmatrix}$

Let $\text{A} = \Re\times\Re$ and$\ast$ be the binary operation on A defined by $\text(a, b) \ast \text{(c, d)} = \text{(a + c, b + d)}. $ Prove that $\ast$ is commutative and associative. Find the identity element for $\ast$ on A. Also write the inverse element of the element (3, – 5) in A.

Evaluate the following integrals:
$\int\frac{\cos^5\text{x}}{\sin\text{x}}\text{ dx}$

Using matrices, solve the following system of equations:
$4x + 3y + 2z = 60.$
$x + 2y + 3z = 45.$
$6x + 2y + 3z = 70.$

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

If the points $(x, -2), (5, 2), (8, 8)$ are collinear, find $x$ using determinants.

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^2=4\text{y}\text{ at }(2,1)$

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x(x + 4)^2$ on $[0,4]$