Differentiate the following functions with respect to x: e^x 2x - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^\text{x}\log\sin2\text{x}$

✓

Answer

Let $\text{y}=\text{e}^\text{x}\log\sin2\text{x}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big[\text{e}^\text{x}\log\sin2\text{x}\big]$
$=\text{e}^\text{x}\frac{\text{d}}{\text{dx}}\log\sin2\text{x}+\log\sin2\text{x}\frac{\text{d}}{\text{dx}}\big(\text{e}^\text{x}\big)$
[Using product rule and chain rule]
$=\text{e}^\text{x}\frac{1}{\sin2\text{x}}\frac{\text{d}}{\text{dx}}(\sin2\text{x})+\log\sin2\text{x}\big(\text{e}^\text{x}\big)$
$=\frac{\text{e}^\text{x}}{\sin2\text{x}}\cos2\text{x}\frac{\text{d}}{\text{dx}}(2\text{x})+\text{e}^\text{x}\log\sin2\text{x}$
$=\frac{2\cos2\text{xe}^\text{x}}{\sin2\text{x}}+\text{e}^\text{x}\log\sin2\text{x}$
$=\text{e}^\text{x}(2\cot2\text{x}+\log\sin2\text{x})$
So,
$\frac{\text{d}}{\text{dx}}\big(\text{e}^\text{x}\log\sin2\text{x}\big)=\text{e}^\text{x}(2\cot2\text{x}+\log\sin2\text{x})$

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

Differentiate $\sin^{-1}\Big\{\frac{2^{\text{x}+1}\times3^\text{x}}{1+(36)^\text{x}}\Big\}$ with respect to x:

If f(x) = x³ + 4x² - x, find f(A), where $\text{A}=\begin{bmatrix}0&1&2\\2&-3&0\\1&-1&0\end{bmatrix}$

Evaluate the following integrals:
$\int\limits_{0}^{\pi}\frac{1}{3+2\sin\text{x}+\cos\text{x}}\text{ dx}$

Find the second order derivatives of the following functions:

$\log(\sin\text{x})$

Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$

Find the points at which the function f given by f(x) = (x - 2)⁴ (x + 1)³ has

local maxima
local minima
point of inflexion

Evaluate the following integrals:
$\int\limits^{8}_2\frac{\sqrt{10-\text{x}}}{\sqrt{\text{x}}+\sqrt{10-\text{x}}}\text{ dx}$

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs. 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Evaluate the following intregals:
$\int\frac{1}{1+3\sin^2\text{x}}\text{ dx}$

Solve the Linear Programming Problem graphically:
Maximize Z = 5x + 3y subject to 3x + 5y $\leq$ 15, 5x + 2y $\leq$ 10, x $\geq$ 0, y $\geq$ 0.