Using the Chain Rule to Differentiate f(g(x^2))

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SUMMARY

The discussion clarifies the application of the chain rule in calculus, specifically for differentiating the function f(g(x^2)). The correct derivative is f'(g(x^2)) * g'(x^2) * 2x, and there is no need to continue differentiating until all instances of x are eliminated. The focus is on differentiating the functions explicitly provided without assuming additional layers of complexity. This reinforces the concept of treating the expression as a double composite function.

PREREQUISITES
  • Understanding of the chain rule in calculus
  • Familiarity with composite functions
  • Basic differentiation techniques
  • Knowledge of function notation and derivatives
NEXT STEPS
  • Review the chain rule in calculus textbooks or online resources
  • Practice differentiating more complex composite functions
  • Explore examples of higher-order derivatives and their applications
  • Learn about implicit differentiation for more advanced scenarios
USEFUL FOR

Students studying calculus, educators teaching differentiation techniques, and anyone looking to strengthen their understanding of the chain rule and composite functions.

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I'm a little confused as to when to stop taking the derivative of the inside function when using the chain rule...

Lets say I have f( g(x^2) )

Would this be correct?
f`( g(x^2) ) * g`(x^2) * 2x ?

Or do I keep on going until the x is completely gone from the equation?
 
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nope, you're done. The idea here is that you only differentiate the functions given. You can't assume that there are still more. All this is is a double composite. It could have been written as F(g(h(x)), which is the same as before except with replcing th x^2 with h(x). Just do the setup and don't second guess yourself.
 

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