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

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SUMMARY

The derivative of cos(t³) is computed using the chain rule, which is essential for differentiating nested functions. The derivative is given by the formula \(\frac{d}{dt}(\cos(t^3)) = -\sin(t^3) \cdot (3t^2)\). The chain rule applies whenever functions are composed, as illustrated by the general formula for nested functions: f(g(h(k(x))))' = f'(g(h(k(x)))) * g'(h(k(x))) * h'(k(x)) * k'(x). Understanding when to apply the chain rule is crucial for accurate differentiation.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically derivatives.
  • Familiarity with the chain rule in differentiation.
  • Knowledge of trigonometric functions and their derivatives.
  • Ability to recognize nested functions in calculus.
NEXT STEPS
  • Study the application of the chain rule in more complex functions.
  • Learn about implicit differentiation and its relationship with the chain rule.
  • Explore the derivatives of other trigonometric functions beyond cos(x).
  • Practice problems involving nested functions to reinforce understanding of the chain rule.
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to improve their skills in differentiation, particularly with nested functions and trigonometric derivatives.

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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?
 
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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)
 
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)
 

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