Find the second - order derivatives of the function e^ 6x 3x - Vidyadip

Question

Find the second $-$ order derivatives of the function $e^{6x }\cos 3x$

✓

Answer

Let $y = e^{6x} \cos \ 3x \therefore \frac{{dy}}{{dx}} = {e^{6x}}.\frac{d}{{dx}}\cos 3x + \cos 3x\frac{d}{{dx}}{e^{6x}}$
$= {e^{6x}}\left( { - \sin 3x} \right)\frac{d}{{dx}}\left( {3x} \right) + \cos 3x.{e^{6x}}\frac{d}{{dx}}\left( {6x} \right)$
$= -{e^{6x}}\sin 3x \times 3 + \cos 3x.{e^{6x}} \times 6$
$= e^{6x}(-3 \sin 3x + 6 \cos 3x)$
$\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = {e^{6x}}\frac{d}{{dx}}\left( { - 3\sin 3x + 6\cos 3x} \right) $
$+ \left( { - 3\sin 3x + 6\cos 3x} \right)\frac{d}{{dx}}{e^{6x}}$
$= {e^{6x}}\left( { - 3\cos 3x \times 3 - 6\sin 3x \times 3} \right) $
$+ \left( { - 3\sin 3x + 6\cos 3x} \right){e^{6x}} \times 6$
$= e^{6x}(-9 \cos 3x - 18 \sin 3x - 18 \sin 3x + 36 \cos 3x)$
$= e^{6x}(27 \cos 3x - 36 \sin3x)$
$= 9e^{6x}(3 \cos 3x - 4 \sin 3x)$

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 "3 Marks Question" from Continuity and Differentiability→Continuity and Differentiability — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Using determinants show that the following points are collinear:
$(5, 5), (-5, 1)$ and $(10, 7)$

Evaluate the following integrals:
$\frac{\text{x}+1}{\text{x}(1+\text{xe}^\text{x})}\ \text{dx}$

If $\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi$ then find the value of $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$

$\text{If}\ \vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},$ find a unit vector parallel to the vector $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}.$

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=2\text{x}^2+3\sin\text{x}\ \text{at}\text{ x}=0$

Integrate the rational function: $\frac{x}{(x-1)(x-2)(x-3)}$

Verify Mean Value Theorem, if $f(x) = x^3 – 5x^2 – 3x$ in the interval $[a, b],$ where $a = 1$ and $b = 3.$ Find all $\text{c}\in(1,\ 3)$ for which $f′(c) = 0.$

A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black?

Classify the following functions as injection, surjection or bijection:$f : R \rightarrow R,$ defined by $f(x) = 1 + x^2$

A box has $5$ blue and $4$ red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?