How do you differentiate #f(x)=cos(7-4x) # using the chain rule?

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Answer 1 · 2016-03-24T01:51:17

#f'(x)=4sin(7-4x)#

The chain rule, when applied to #cos(x)#, states that

#d/dxcos(u)=-sin(u)*u'#

This is very similar to the typical differentiation for #cos(x)# without the chain rule:

#d/dxcos(x)=-sin(x)#

except for that when the chain rule is applied the derivative of the function #u# is also multiplied to the #-sin(u)# expression.

Applying this to #f(x)=cos(7-4x)#, where #u=7-4x#, we see that

#f'(x)=-sin(7-4x)*d/dx(7-4x)#

Note that the derivative of #7-4x# is #-4#. This simplifies our overall derivative expression:

#f'(x)=-sin(7-4x)*(-4)#

#f'(x)=4sin(7-4x)#