Chapter 14: Derivatives

Recall,

$f'(x) = \lim {h \rightarrow 0}_{f(x+h) - f(x) }{(x+h)-x} = \frac{rise}{run}$

$=$ "The derivative of $f$ at $x$"

$=$ "The slope of $f$ at $x$"

Example

Find the slope of $f(x) = \frac{1}{2x^2}$ at $x=3$

Apply the definition of slope

$f'(3) = \lim_{h \rightarrow c} \frac{f(3+h) - f(3)}{(3+h) - 3}$

$= \lim_{h \rightarrow 0} \frac{ f(3+h)-f(3)}{h}$

$= \lim_{h \rightarrow 0} \frac{ \frac{1}{2}(3+h)2- \frac{1}{2}(3)2 }{ h}$

$= \lim_{h \rightarrow 0} \frac{ \frac{1}{2}(3^2+2 \cdot 3 \cdot h+h^2)2- \frac{1}{2}(3)2 }{ h}$

$= \lim_{h \rightarrow 0} \frac{ \frac{1}{2}(6h+h^2) }{ h}$

$= \lim_{h \rightarrow 0} \frac{1}{2} \cdot 6 + \frac{1}{2} h$

$=3$

Thus, the slope at $x=3$ is $f'(3)=3$

Rate of Change

The physical interpretation of derivatives is that they measure rates of change.

Definition

If the distance to an object at time $t$ is $d(f)$ then its:

VELOCITY $=d'(+) =s(t)$

SPEED $=|d'(t)| \rightarrow$ Speed ignores direction

ACCELERATION $= s'(f)$

Discuss

An object dropped near Earth travels $d(t) = 4.9t^2$ meters in $t$ seconds. How far do you need to drop an object so that it hits the ground at $9.8m/s$?

Find speed at time $t$

$d'(t) =9.8t$

Solve for $d'(t) =9.8$

$d'(t) = 9.8t =9.8 \Rightarrow1$

Find the distance traveled at $t =1$

$d(1)=4.9t^2 =4.9m$

Thus, we must drop the object from a height of 4.9m.

ExamplE

Gregor Mendel discovered the laws of ''genetic inheritance'' by studying populations of peas.

He found: If $p$ is the percentage of smooth skinned peas in a population then the proportion in the next generation will be:

$s= 2p(1-p)=2p-2p^2$

When does $s$ change most as a function of $p$?

Find $\frac {ds}{dp}$

$\frac{ds}{dp} = \lim_{h \rightarrow 0} \frac{s(p+h) -s(p)}{(p+h)-p}$

$= \lim_{h \rightarrow 0} \frac {s(p+h)-s(p)}{h}$

$= \lim_{h \rightarrow 0} \frac {2(p+h)-2(p+h)^2]-[2p-2p^2]}{h}$

$= \lim_{h \rightarrow 0} \frac {2h-4ph-2h^2}{h}$

$= \lim_{h \rightarrow 0} 2-4p-2h$

$=2-4p$

Thus, the proportion changes most when $p=0$

That is, introducing a few smooth skin peas will have the largest affect on the population

We need a more systematic way of calculating derivatives.

Differentiation Rules

Fact

If $f(x) =k$ for all $x$ then $f'(x)=0$

Discuss

Why is this true?

$f'(x) = \lim_{h \rightarrow 0 } \frac{f(x+h)-f(x)}{h}$ $\rightarrow$ Substitute $f(x)=k$

$= \lim_{h \rightarrow 0 } \frac{k-k}{h}$

$\frac{d}{dx}(k) =0$

Fact

If $f(x)= c \cdot g(x)$ for some number $c$ then $f'(x) = c \cdot g'(x)$

Discuss

Why is this true ? $\rightarrow$ Transformation

$f'(x) = \lim_{h \rightarrow 0} \frac { f(x+h)-f(x)}{h}$ $\rightarrow$ Substitute $f(x) = c \cdot g(x)$

$= \lim_{h \rightarrow 0} \frac {c \cdot g(x+h)-c \cdot g(x)}{h}$

$=c \lim_{h \rightarrow 0} \frac{g(x+h)-g(x)}{h }$ $\rightarrow$ Limit Law

$=cg'(x)$

$\frac{d}{dx}[c \cdot g(x)] =c \frac {d}{dx}[g(x)]$

Summary

Constants $\rightarrow$ $f(x)= k \Rightarrow f'(x)=0$

Scaling $\rightarrow$ $f(x)=c \cdot g(x) \Rightarrow f'(x)= c \cdot g'(x)$