Derivatives and the chain rule

In summary, the conversation discusses different approaches to finding the derivative of a function, h, and how they yield different results. One approach uses the chain rule, while the other only looks at the values of f and g. The speaker also clarifies the use of prime notation in differentiating functions.
  • #1
AL107
3
0
Homework Statement
x: -1 1 3
f(x): 6 3. 1
f’(x): 5. -3 -2
g(x): 3. -1. 2
g’(x): -2. 2. 3

The table above gives values of f, f', g, and g' at selected values of x. If h(x) = f(g(x)), then h'(1) =
(A) 5
(B) 6
(C) 9
(D) 10
(E) 12
Relevant Equations
h(x)=f(g(x))
I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they yield different results. Can someone clarify? Thank you!
 
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  • #2
Oh, here is a better table:
FE83F39F-47E7-4F3D-A38F-7C8208306204.jpeg
 
  • #3
AL107 said:
An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table.
How can that possibly be right? The chain rule applies.
 
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  • #4
PeroK said:
How can that possibly be right? The chain rule applies.
Thank you!
 
  • #5
In the second method, you are differentiating h by x and f by g, so you are not performing the same operation on both sides of the equation. Be careful when using the prime notation: h'(x) means dh/dx, but f'(g) means df/dg. It may be helpful to write out the derivatives explicitly:
h(x) = f(g)
dh/dx = df/dx = df/dg*dg/dx
 
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  • #6
Intuitively, you can not just look at f' because x has to go through g before f is applied. Consider the simple example, f(x)=x. Then h(x)=f(g(x)) = g(x) and clearly h'= g', so g' can not be ignored.
 
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FAQ: Derivatives and the chain rule

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It measures how much a function changes in response to a small change in its input.

2. How is the chain rule used to find derivatives?

The chain rule is a rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function times the derivative of the inner function.

3. Why is the chain rule important?

The chain rule is important because it allows us to find the derivatives of more complex functions by breaking them down into simpler functions. It is a fundamental tool in calculus and is used in many real-world applications, such as in physics, economics, and engineering.

4. Can the chain rule be applied to any type of function?

Yes, the chain rule can be applied to any type of function, as long as it is a composite function. This means that the function is made up of two or more functions nested inside each other.

5. How do you know when to use the chain rule?

You should use the chain rule whenever you have a composite function and need to find its derivative. This includes functions that involve trigonometric, exponential, or logarithmic functions, as well as functions that are composed of multiple functions.

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