Calculating Derivatives of Composite Functions

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SUMMARY

The discussion focuses on calculating derivatives of composite functions using provided values for functions F and G. Specifically, it addresses the evaluation of H(4) and H'(4) for various forms of H, including H(x)=F(G(x)), H(x)=G(F(x)), and H(x)=F(x)/G(x). The participants confirm that the chain rule applies for problems B and D, while the quotient rule is necessary for problem E. The correct application of these rules leads to accurate derivative calculations.

PREREQUISITES
  • Understanding of composite functions and their derivatives
  • Proficiency in the chain rule for differentiation
  • Knowledge of the quotient rule for differentiation
  • Familiarity with evaluating functions and their derivatives at specific points
NEXT STEPS
  • Study the chain rule in detail, including examples and applications
  • Learn the quotient rule and practice problems involving it
  • Explore the product rule for differentiation and its applications
  • Review composite functions and their derivatives in calculus textbooks
USEFUL FOR

Students studying calculus, particularly those learning about derivatives of composite functions, and educators seeking to clarify these concepts for their students.

kxpatel29
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Homework Statement


Given F(2)=5, F'(2)=6, F(4)=3, F'(4)=6 and G(3)=6, G'(3)=3, G(4)=2, G'(4)=1
A. H(4) if H(x)=F(G(x)) = ?
B. H'(4) if H(x)=F(G(x)) = ?
C. H(4) if H(x)=G(F(x)) = ?
D. H'(4) if H(x)=G(F(x)) =?
E. H'(4) if H(x)=F(x)/G(x) =?


Homework Equations


My teacher did not teach this yet, so I have no idea. My book doesn't even cover this


The Attempt at a Solution


Would we just use the given values? For E, F(x)/G(x)=1/2?
 
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kxpatel29 said:

Homework Statement


Given F(2)=5, F'(2)=6, F(4)=3, F'(4)=6 and G(3)=6, G'(3)=3, G(4)=2, G'(4)=1
A. H(4) if H(x)=F(G(x)) = ?
B. H'(4) if H(x)=F(G(x)) = ?
C. H(4) if H(x)=G(F(x)) = ?
D. H'(4) if H(x)=G(F(x)) =?
E. H'(4) if H(x)=F(x)/G(x) =?


Homework Equations


My teacher did not teach this yet, so I have no idea. My book doesn't even cover this


The Attempt at a Solution


Would we just use the given values? For E, F(x)/G(x)=1/2?
You have to use the given values, yes, but except for A and C, you need to take the derivative using either the chain rule (B and D) or quotient rule (E).

For example, if problem E were changed like so:
Given F(2)=5, F'(2)=6, F(4)=3, F'(4)=6 and G(3)=6, G'(3)=3, G(4)=2, G'(4)=1
Find H'(4) if H(x)=F(x)G(x)

then H'(4) = F(4)G'(4) + F'(4)G(4) = 3*1 + 6*2 = 3 + 12 = 15.
Here I have used the product rule.

Similar idea for all of them.
 

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