How do you differentiate $f (x) = e^{1 + 4 x} \cdot sin (5 - x)$ $f(x)=e^(1+4x)*sin(5-x)$ using the product rule?

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Answer 1 · 2016-10-02T15:43:28

$f'(x)=e^(1+4x){4sin(5-x)-cos(5-x)}$ .

Let, $f(x)=e^(1+4x)sin(5-x)=uv, say, where, u=e^(1+4x), &, v=sin(5-x).$

Using Product Rule, $f'(x)=u(dv)/dx+v(du)/dx............(star)$

$u=e^(1+4x) rArr (du)/dx=e^(1+4x)*d/dx(1+4x)..........."[Chain Rule]"$

$:. (du)/dx=4e^(1+4x)............(1)$ .

$v=sin(5-x) rArr (dv)/dx=cos(5-x)d/dx(5-x)......"[Chain Rule]"$

$:. (dv)/dx=-cos(5-x)...................(2)$

Using $(1) & (2)" in "(star),$

$f'(x)=4e^(1+4x)sin(5-x)-cos(5-x)e^(1+4x)$ .

$:. f'(x)=e^(1+4x){4sin(5-x)-cos(5-x)}$ .

How do you differentiate f(x)=e^(1+4x)*sin(5-x)f(x)=e1+4x⋅sin(5−x)f(x)=e^(1+4x)*sin(5-x) using the product rule?