Question

Differentiate the following functions using product rule.​

Answer 1

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)}`