Download App

Higher Order Derivatives — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSHigher Order Derivatives3 Marks
Question
If $\text{y}=\text{e}^\text{x}\cos\text{x},$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\text{e}^\text{x}\cos(\text{x}+\frac{\pi}{2}).$
✓

Answer

Here
$\text{y}=\text{e}^\text{x}\cos\text{x}$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}\cos\text{x}-\text{e}^\text{x}\sin\text{x}=\text{e}^\text{x}(\cos\text{x}-\sin\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}=\text{e}^\text{x}(\cos\text{x}-\sin\text{x})+\text{e}^\text{x}(-\sin\text{x}-\cos\text{x})$
$=\text{e}^\text{x}\cos\text{x}-\text{e}^\text{x}\sin\text{x}-\text{e}^\text{x}\sin\text{x}-\text{e}^\text{x}\cos\text{x}$
$=-2\text{e}^\text{x}\sin\text{x}$
$=2\text{e}^\text{x}\cos(\text{x}+\frac{\pi}{2})$

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 "3 Marks Question" from Higher Order Derivatives→Higher Order Derivatives — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the area of the triangle with vertices at the points: $(0, 0), (6, 0)$ and $(4, 3)$
→
Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big)+\sec^{-1}\Big(\frac{1+\text{x}^2}{1-\text{x}^2}\Big),\text{x}\in\text{R}$
→
Solve the following differential equation:
$(y^2 - x^2) dy = 3xy\ dx.$
→
Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
→
Check the commutativity and associativity of the following binary operations: $'^*\ ' $on $Q$ defined by $a * b = ab^2$ for all $a, b \in Q.$
→
For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?
→
If $[\cdot]$ and $\{\cdot\}$ denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_0\sin\{\text{x}\}\text{dx}$ 
→
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
  1. An injective map from A to B.
  2. A mapping from A to B which is not injective.
  3. A mapping from A to B.
→
Integrate the function $\sqrt{1-4 x-x^{2}}$
→
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.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$
→