Notes
Basic rules
- Power rule: $(x^n)' = nx^{n-1}$. Constant multiple and sum rules let you differentiate term by term. The derivative of a constant is 0.
- Product rule: $(fg)' = f'g + fg'$.
- Quotient rule: $\displaystyle \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$.
Chain rule
- $[f(g(x))]' = f'(g(x))\,g'(x)$: differentiate the outside, then multiply by the derivative of the inside.
Special functions
- $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \sec^2 x$.
- $(e^x)' = e^x$, $\displaystyle (\ln x)' = \frac{1}{x}$, $(a^x)' = a^x \ln a$.
- Chain versions: $(e^u)' = e^u u'$, $\displaystyle (\ln u)' = \frac{u'}{u}$, $(\sin u)' = u'\cos u$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Power rule on each term; the derivative of 2 is 0.
Product rule: $(2x)\sin x + x^2 \cos x$.
Quotient rule: $\displaystyle \frac{e^x \cdot x - e^x \cdot 1}{x^2}$.
Outside: $-\sin(5x)$. Inside derivative: 5.
$\displaystyle \frac{u'}{u}$ with $u = x^2 + 1$.
$y' = 3(x^2 + 1)^2(2x)$. At $x = 1$: $3(4)(2) = 24$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $g(x) = x^3 \sin x$, $h(x) = (2x + 1)^4$ and $k(x) = \frac{x}{x^2 + 1}$.
(a) Find $g'(x)$.
Show the answer and scoring guide
Product rule: $g'(x) = 3x^2 \sin x + x^3 \cos x$.
Check off the points you earned:
(b) Find $h'(x)$.
Show the answer and scoring guide
Chain rule: $h'(x) = 4(2x + 1)^3 \cdot 2 = 8(2x + 1)^3$.
Check off the points you earned:
(c) Find $k'(x)$ and show that $k$ has a horizontal tangent at $x = 1$.
Show the answer and scoring guide
Quotient rule: $\displaystyle k'(x) = \frac{(1)(x^2 + 1) - x(2x)}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$. At $x = 1$ the numerator is 0, so $k'(1) = 0$ and the tangent is horizontal.
Check off the points you earned: