The Chain Rule
Warmup Let us start with a quick quiz on using the "Calculation Thought Experiment (CTE)" discussed in the preceding tutorial:
Calculation Thought Experiment (CTE)
The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, or difference:
Using the Calculation Thought Experiment (CTE) to Differentiate a Function
Examples
1. (3x2-4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product. 2. (2x-1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient. 3. x2 + (4x-1)(x+2) can be computed by first calculating x2, then calculating the product (4x-1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum. 4. (3x2-1)5 can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power. |
Q Now that we are done with the preliminaries, what is the chain rule?
A Here is an example: We know that the derivative of x3 is 3x2. What, then, would you say is the deriviative of something more complicated raised top the third power, for instance (2x + x-1.4)3 ?
Q Is it not just 3(2 + 1.4x0.4)2 ?
A No. To find the correct answer, we use the chain rule.
The Chain Rule
If u is a differentiable function of x, and f is a differentiable function of u, then f is a differentiable function of x, and:
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Example
Taking f(x) = x3, we get
In words:
This is sometimes referred to as an example of the generalied power rule. Quick Examples
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Q Why is the chain rule true?
A Press here for a proof.
Here are some for you to try. Note Use proper graphing calculator format to input your answers (spaces are ignored).
Q What about things other than powers -- things such as ln(x2 + 4x), for instance?
A The following table shows how we apply the chain rules to a whole variety of functions, but the derivatives of these functions appear later, so have patience, or follow the links below.
(Chain Rule) |
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General form of Chain Rule | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Generalized Power Rule | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Power rule with n = 1/2 |
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See next tutorial. |
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Take me to text on trig functions! |
\frac{d}{dx} = ? \frac{d}{dx} = ?
Try some exercises for this topic here. Alternatively, try some of the exercises in Section 4.4 in Applied Calculus or Section 11.4 in Finite Mathematics and Applied Calculus. |