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. (3x^{2}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. (2x1)/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. x^{2} + (4x1)(x+2) can be computed by first calculating x^{2}, then calculating the product (4x1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum. 4. (3x^{2}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 x^{3} is 3x^{2}. 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 + 1.4x^{0.4})^{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:


Example
Taking f(x) = x^{3}, we get
In words:
This is sometimes referred to as an example of the generalied power rule. Quick Examples

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(x^{2} + 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) 



General form of Chain Rule  



Generalized Power Rule  



Power rule with n = 1/2  



See next tutorial.  



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.