Differentiate the following functions with respect to x: e^ 3x 2x - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$

✓

Answer

Consider $\text{y}=\text{e}^{3\text{x}}\cos2\text{x}$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\text{e}^{3\text{x}}\cos2\text{x}$
$=\text{e}^{3\text{x}}\times\frac{\text{d}}{\text{dx}}(\cos2\text{x})+\cos2\text{x}\frac{\text{d}}{\text{dx}}(\text{e}^{3\text{x}})$
[Using chain rule]
$=\text{e}^{3\text{x}}\times(-\sin2\text{x})\frac{\text{d}}{\text{dx}}(2\text{x})+\cos2\text{xe}^{3\text{x}}\frac{\text{d}}{\text{dx}}(3\text{x})$
[Using chain rule]
$=-2\text{e}^{3\text{x}}\sin2\text{x}+3\text{e}^{3\text{x}}\cos2\text{x}$
$=\text{e}^{3\text{x}}(3\cos2\text{x}-2\sin2\text{x})$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}(\text{e}^{3\text{x}}\cos2\text{x})=\text{e}^{3\text{x}}(3\cos2\text{x}-2\sin2\text{x})$

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 Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²) dx = (y³ – 3x²y) dy, where C is a parameter.

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) = x² - 2x + 4 on [1, 5]

Maximise Z = 3x + 4y
subject to the constraints: $\text{x}+\text{y}\leq4,\text{x}\geq0,\ \text{y}\geq0.$

Find the second order derivatives of the following functions:
$\text{y}=\tan^{-1}\text{x}$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=1-\text{x + y}-\text{xy}$

Let $\text{A} = \text{R} - \left\{3\right\}, \text{B} = \text{R} - \left\{1\right\}. \text{Let f : A} \rightarrow \text{B}$ be defined by $\text{f(x)}\frac{\text{x - 2}}{\text{x - 3}}, \forall \text{ x} \in \text{A}.$ Show that f is bijective. Also, find

$\text{x, if f}^{-1} \text{(x) = 4}$
$\text{f}^{-1} (7)$

$\int\limits_{0}^{\pi}\text{x}\log\sin\text{x dx}$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\frac{3}{2}}\text{x}}{\sin^{\frac{3}{2}}\text{x}+\cos^{\frac{3}{2}}\text{x}}\text{ dx}$

Find the area of the region bounded by the ellipse $\frac{\text{x}^2}{4}+\frac{\text{y}^2}{9}=1.$

Evaluate the following integrals:

$\int\text{x}^2\sin^{-1}\text{x dx}$