Differentiation (Pure 2)
Standard derivatives
$$\frac{d}{dx}e^x = e^x, \quad \frac{d}{dx}\ln x = \frac{1}{x}, \quad \frac{d}{dx}\sin x = \cos x,$$
$$\frac{d}{dx}\cos x = -\sin x, \quad \frac{d}{dx}\tan x = \sec^2 x.$$
Practice
The derivative of ln x is 1/x. What is its value at x = 4?
d/dx(ln x) = 1/x = 1/4 = 0.25.
Practice
The derivative of eˣ is eˣ. What is its value at x = 0?
d/dx(eˣ) = eˣ, and e⁰ = 1.
Product, quotient, parametric
- product rule: $(uv)' = u'v + uv'$.
- quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.
- parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.
- implicit: differentiate every term in $x$, chain-rule the $y$ terms, then solve for $\dfrac{dy}{dx}$.
Practice
The product rule says (uv)′ equals:
The product rule: (uv)′ = u′v + uv′.
You've got it
Key idea
- learn: $\dfrac{d}{dx}e^x = e^x$, $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$, $\dfrac{d}{dx}\sin x = \cos x$, $\dfrac{d}{dx}\cos x = -\sin x$
- product rule $(uv)' = u'v + uv'$; quotient rule $\dfrac{u'v - uv'}{v^2}$
- parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$