Question

If f(x) = e^x sin x, then df(x) /dx
is:-
1. e^x sin x
2.e^x cos x
3.e^x(sin x + cos x)
4. e^x(sin x - cosx) ​

Answer 1

Answer:

2. e^x cos x

The answer is option 2.

df(x)/dx = d e^x sinx / dx

= e^x cosx

(differentiation of e^x is e^x and differentiation of sinx is cos x)

hope it helps :)

Answer 2

Answer:

$\boxed{\mathfrak{4. \ e^x(sin \ x + cos \ x)}}$

Given:

$\rm f(x) = {e}^{x} \: sin \: x$

Explanation:

$\rm \implies \dfrac{d}{dx}f(x) \\ \\ \rm \implies \dfrac{d}{dx} ( {e}^{x} \: sin \: x) \\ \\ \rm By \: using \: the \: product \: rule, \\ \rm \frac{d}{dx} (uv)= v \frac{du}{dx} + u \frac{dv}{dx} : \\ \\ \rm \implies {e}^{x} ( \frac{d}{dx} sin \: x) + sin \: x (\frac{d}{dx} {e}^{x} ) \\ \\ \rm \implies {e}^{x} \: cos \: x + sin \: x (\frac{d}{dx} {e}^{x} ) \\ \\ \rm \implies {e}^{x} \: cos \: x + {e}^{x} \: sin \: x \\ \\ \rm \implies {e}^{x} ( sin \: x + cos \: x)$