Is the chain rule applied in the derivative of cos(t^3) ?

[engstudent363]

How would you compute the derivative of cos(t^3)? Would you use the chain rule? Does anyone have a good way of recognizing when to use chain rule and when not to?

 

[quasar987]

You use the chain rule because it is useful. For instance, here I don't know what the derivative of cos(t³) is. But I know what the derivative of cos(x) is (-sin(x)) and what the derivative of t³ is (3t²). So I choose to make use of the chain rule and say, since cos(t³) is the composition of x-->cos(x) and t-->t³, then

$$\frac{d}{dt}(\cos(t^3))=-\sin(t^3)\cdot (3 t^2)$$

 

[ice109]

you use the chain rule anytime you have nested functions. hence if you have a function:

f( g( h( k(x) ) ) ) it's derivative with respect to x is f'( g( h( k(x) ) ) ) * g'( h( k(x) ) ) * h'( k(x) ) * k'(x)