3.5 Differentiating Compositions of Functions With the Chain Rule

2. 3.5Differentiating Compositionsof FunctionsWith the Chain Rule

3. Suppose you want to differentiate a composite function: The rules we know (power, product, and quotient) don’t really apply here! Our tool is the Chain Rule. Thinking of rates of change, You’ve probably used something like the above in a science class. Suppose you’re making a road trip, and you plan to stop every 100 miles for a coke. If you’re traveling at 60 miles per hour, then your rate of soda consumption is:

4. The Chain Rule: If f and g are both differentiable and is the composite function defined by then In Liebniz notation, this looks like: Remember that these are limits, not actual quotients so we can’t just ‘cancel’ du . Example:Differentiate f(u) = f’(u) = g(x) = g’(x) =

5. Let’s try it two ways: Chain Rule: Simplify & Power Rule

6. This is different from the order of operations: we’re working from the OUTSIDE, Not the inside!

7. Example: Differentiate Differentiate:

8. Power Rule Combined with the Chain Rule: If n is any real number and u=g(x) Is differentiable, then: Example: Differentiate

9. Differentiate:

10. Differentiate:

11. Derive a rule for differentiating exponential function having base a:

13. Read 3.5 p 217 – 224 Work p 224 # 1 - 4, 6 - 8, 10, 11, 14, 16, 17, 23 & 26