Differentiate the following functions using product rule.

Question

asked Jan 19, 2023 58.8k views

1 Answer

Wkrueger · Answer 1 · 2023-01-25T18:50:19+0000

d) By the product rule,

$y = 5 e^x \sin(x) \implies (dy)/(dx) = (d5)/(dx) e^x \sin(x) + 5 (de^x)/(dx) \sin(x) + 5 e^x (d\sin(x))/(dx)$

The derivative of a constant is 0; the derivative of
e^x is
e^x ; the derivative of
$\sin(x)$ is
$\cos(x)$ . So

$(dy)/(dx) = \boxed{5 e^x \sin(x) + 5 e^x \cos(x)}$

e) Rewrite
$\sqrt x = x^(1/2)$ . By the product rule,

$y = \sqrt x\,e^x = x^(1/2) e^x \implies (dy)/(dx) = (dx^(1/2))/(dx) e^x + x^(1/2) (de^x)/(dx)$

By the power rule,

$(dx^(1/2))/(dx) = \frac12 x^(1/2-1) = \frac12 x^(-1/2)$

Then

$(dy)/(dx) = \boxed{(e^x)/(2\sqrt x) + \sqrt x\,e^x}$

f) By the product rule,

$y = (1 + \sin(x)) \tan(x) \implies (dy)/(dx) = (d(1+\sin(x)))/(dx) \tan(x) + (1+\sin(x)) (d\tan(x))/(dx)$

The derivative of
$\tan(x)$ is
$\sec^2(x)$ . So

$(dy)/(dx) = \cos(x)\tan(x) + (1 + \sin(x)) \sec^2(x)$

which we can simplify using

$\tan(x) = (\sin(x))/(\cos(x))$

and expand to get

$(dy)/(dx) = \boxed{\sin(x) + \sec^2(x) + \tan(x) \sec(x)}$