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" 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

Let f, g, h be real functions given by f(x) = sinx, g(x) = 2x and h(x) = cosx. Prove that fog = go(fh).

Evaluate the following integrals:

$\int\frac{\text{x}\cos^{-1}\text{x}}{\sqrt{1-\text{x}^3}}\text{dx}$

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}-1&4\\2&3 \end{vmatrix}$

If $\text{y}=\log(\sin\text{x})$ Prove that $\frac{\text{d}^3\text{y}}{\text{dx}^3}=2\cos\text{x}\ \text{cosec}^3\text{x}$

Evaluate the following integrals:
$\int\sin^3\text{x}\cos^6\text{x}\text{ dx}$

Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$

Show that the four points having position vectors $6\hat{\text{i}}-7\hat{\text{j}},\ 16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},\ 3\hat{\text{j}}-6\hat{\text{k}},\ 2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

Express the following matrices as the sum of a symmentric and a skew symmentric matrix:

$\begin{bmatrix}3&5\\1&-1\end{bmatrix}$

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

x_i	-1	0	1	2	3
p_i	0.3	0.1	0.1	0.3	0.2

If $\text{y}=\cos^{-1}\text{x},$ Find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ in terms of y alone.