The Chain Rule is used for differentiating composite functions. The rule itself looks really quite simple (and it is not too difficult to use). The most important thing to understand is when to use it and then get lots of practice. It is useful to get fluent in applying The Chain Rule, as this will save you lots of time in an exam. This is especially true since some functions are composed of more than two functions and The Chain Rule is often used as part of a bigger question that use other rules ( Product and Quotient Rule ) or questions about Related Rates of Change or Implicit Differentiation.
Key Concepts
On this page, you should learn about
differentiating composite functions using the chain rule
Essentials
The following videos will help you understand all the concepts from this page
Why and How
The following video explains why we need The Chain Rule and how to use it. There will be two examples in which we will find the derivative of \((3x+1)^4\) and \(e^{3x^2}\)
Differentiating [f(x)]n
In the following video, we will look for some simple patterns so that we can quickly differentiate functions in the form [f(x)]n
Differentiating ef(x) and ln[f(x)]
In the following video, we will look for some simple patterns so that we can quickly differentiate functions in the form ef(x) and ln[f(x)]
Consider two functions f and g and their derivatives f' and g'. The following table shows the values for the two functions and their derivatives at x = -1, 0, 1
-1
0
1
f(x)
-2
1
6
g(x)
3
-1
1.5
f'(x)
2
3
4
g'(x)
4
-2
3
\(h(x) = f\circ g(x)\)
Find h'(0)
h'(0) =
h'(x) = g'(x)f'(g(x))
h'(0) = g'(0)f'(g(0))
h'(0) = g'(0)f'(-1)
h'(0) = -1\(\times\)2
h'(0) = -2
MY PROGRESS
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