If y=e^ -x , show that d^2y dx^2 =2e^ -x . - Vidyadip

Question

If $\text{y}=\text{e}^{-\text{x}}\cos\text{x},$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\text{e}^{-\text{x}}\sin\text{x}.$

✓

Answer

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

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 "3 Marks Question" 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

Find the angle between the line $\vec{\text{r}}=(2\hat{\text{i}}+3\hat{\text{j}}+9\hat{\text{k}})+\lambda(2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})$ and the plane $\vec{\text{r}}\cdot(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})=5$

Find which of the binary operations are commutative and which are associative.
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)

Find the area of the triangle with vertices at the points:
$(0, 0), (6, 0)$ and $(4, 3)$

Show that the lines $\frac{\text{x}-5}{7}=\frac{\text{y}+2}{-5}=\frac{\text{z}}{1}\ \text{and}\ \frac{\text{x}}{1}=\frac{\text{y}}{2}=\frac{\text{z}}{3}$ are perpendicular to each other.

Find two numbers whose sum is $24$ and whose product is as large as possible.

A vector $\vec{\text{r}}$ is inclined to x-axis at 45º and y-axis at 60º. If $|\vec{\text{r}}|=8$ units, find $\vec{\text{r}}$.

Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by:
$f(x) = x^2 + 2x - 3$ and $g(x) = 3x - 4$

State Lagrange's mean value theorem.

Find the mean and standard deviation of the following probability distributions:

$x_i$	1	2	3	4
$p_i$	0.4	0.1	0.2	0.3

Evaluate the following integrals:
$\int_{0}^\limits{\frac{1}{2}}\frac{\text{x}\sin^{-1}\text{x}}{\sqrt{1-\text{x}^2}}\text{ dx}$