How do you differentiate #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)}#.