How do you differentiate #f(x)=cose^(4x)# using the chain rule.?

1 Answer
Dec 7, 2015

Answer:

# f'(x) = - 4 sin (e^(4x)) * e^(4x) #

Explanation:

Your chain can be defined as follows:

#f(x) = color(green)(cos) color(blue)(e^(color(red)(4x)))#

#color(white)(xxx)= cos e^(w(x)) color(white)(xxx) " where " w(x) = color(red)(4x)#

#color(white)(xxx)= cos v(w) color(white)(xxxiii) " where " v(w) = color(blue)(e^w)#

#color(white)(xxx)= u(v) color(white)(xxxxxxi) " where " u(v) = color(green)(cos(v))#

Thus, to compute the derivative, you need to build the derivatives of #w(x)#, #v(w)# and #u(v)#:

#w(x) = 4x color(white)(xxx) => color(white)(xx) w'(x) = 4#

#v(w) = e^w color(white)(xxx) => color(white)(xx) v'(w) = e^w = e^(4x)#

#u(v) = cos v color(white)(xx) => color(white)(xx) u'(v) = - sin v = - sin(e^w) = - sin(e^(4x))#

Now, the only thing left to do is to multiply the three derivatives!

#f'(x) = u'(v) * v'(w) * w'(x) #
# color(white)(xxxx) = - sin (e^(4x)) * e^(4x) * 4#
# color(white)(xxxx) = - 4 sin (e^(4x)) * e^(4x) #