Section 2.5, the chain rule
The Chain Rule
My motto, which you’ll hear me say again and again: “it’s always the chain rule.”
We can differentiate polynomials, trigonometric functions, rational (fractional) functions, etc… but what about functions that are compositions of multiple functions?
The chain rule:
If $f$ and $g$ are differentiable functions and $y = f(g(x))$, then:
$$ \frac{d}{dx} (f(g(x)) = f’(g(x)) g’(x) $$
Demo example
Find the derivative of $(3x+1)^2$ in two ways. First by using the chain rule, then by expanding it and differentiating the polynomial.
Examples
- Find the derivative of $y = \sin(x^2)$
- Find the derivative of $y=\sin^2 (x)$
- Find $\dfrac{d}{dx} \sqrt{3x^2 - 4x}$
- Find $\dfrac{dy}{dx}$ where $\displaystyle y = \left(\frac{t}{1-2t}\right)^3$
Super examples!
- Find $k’(t)$ if $k(x) = \sin(\cot (\sec x))$.
- Find the derivative of $f(x) = \sin^2(\tan (x^2))$
Now practice!
Head over to WebAssign and work on the exercises.
Have a question? A comment? Want to clarify something above or on your homework set? Let me know!