exercise - basic differentiation concepts

worked example: let A and B be points on the function $f(x)=-x^2+x-2$ at $x=1$ and $x=1+h$ respectively. find the gradient of line AB in terms of $h$.

let’s draw something

to find the gradient of line AB, we need $\frac{f(1+h)-f(1)}{h}$

$=\frac{-(1+h{)}^2+(1+h)-2-(-1^2+1-2)}{h}$
$=\frac{-(h^2+2h+1)+1+h-2+1-1+2}{h}$
$=\frac{-h^2-h}{h}$
$=\frac{h(-h-1)}{h}$
$=-h-1$

worked example: A secant line intersects the curve $y=x^2+5x$ at two points that have x-coordinates $2$ and $2+\Delta x$. Express the slope of the secant line in terms of $\Delta x$

figure: Clumsy graph here haha

the slope of the secant line here is rise over run, which is $\frac{f(2+\Delta x)-f(2)}{\Delta x}$
$=\frac{(2+\Delta x{)}^2+5(2+\Delta x)-(2^2+5(2))}{\Delta x}$
$=\frac{\Delta x^2+4\Delta x+4+10+5\Delta x-14}{\Delta x}$
$=\frac{\Delta x^2+9\Delta x}{\Delta x}$
$=\Delta x+9$

worked example: What is the slope of the secant line that intersects the graph of $f(x)=0.5^{-x}$ at $x=1$ and $x=5$?

the slope of the secant line is rise over run, which is $\frac{0.5^{-5}-0.5^{-1}}{5-1}$
$=\frac{32-2}{4}$
$=\frac{15}{2}$

worked example: let A and B be points on the function $f(x)=x^2+5x+2$ at $x=8$ and $x=8+h$ respectively. find the gradient of line AB in terms of $h$.

let’s draw something

to find the gradient of line AB, we need $\frac{f(8+h)-f(8)}{h}$

$=\frac{h^2+16h+64+40+5h+2-(8^2+40+2)}{h}$
$=\frac{h^2+21h}{h}$
$=h+21$

worked example: A secant line intersects the curve $y=-2x^2-7x$ at two points where $x=-4$ and $x=t$. What is the slope of the secant line in terms of $t$?

The slope is rise over run, $\frac{-2t^2-7t-(-2(-4{)}^2+28)}{t-(-4)}$
$=\frac{-2t^2-7t+4}{t+4}$
$=\frac{-2t^2-8t+t+4}{t+4}$
$=\frac{-2t(t+4)+1(t+4)}{t+4}$
$=-2t+1$

worked example: Differentiate $f(x)=2x^2-\frac{1}{x}$ using First Principles

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{2(x+h{)}^2-\frac{1}{x+h}-(2x^2-\frac{1}{x})}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{2(x^2+2xh+h^2)-\frac{1}{x+h}-2x^2+\frac{1}{x}}{h}$
$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{4xh+2h^2+\frac{-x+x+h}{x(x+h)}}{h}$
$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{4xh+2h^2+\frac{h}{x^2+hx}}{h}$
$f^{\prime }(x)=\underset{h\to 0}{\lim }4x+2h+\frac{1}{x^2+hx}$
$f^{\prime }(x)=4x+\frac{1}{x^2}$

worked example: Differentiate $f(x)=\frac{1}{x+2}$ using First Principles

$f^{\prime }(x)\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\frac{1}{x+h+2}-\frac{1}{x+2}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }(\frac{1}{h})(\frac{1}{x+h+2}-\frac{1}{x+2})$

$f^{\prime }(x)=\underset{h\to 0}{\lim }(\frac{1}{h})\frac{x+2-x-h-2}{(x+2)(x+h+2)}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }(\frac{1}{h})\frac{-h}{(x+2)(x+h+2)}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{-1}{(x+2)(x+h+2)}$

$f^{\prime }(x)=\frac{-1}{(x+2{)}^2}$

worked example: Use First Principles to find the derivative of $f(x)=x^2$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^2-(x{)}^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{x^2+2xh+h^2-x^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h(2x+h)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{1(2x+h)}{1}$

$f^{\prime }(x)=2x$

worked example: Use First Principles to find the derivative of $f(x)=\frac{1}{x}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{1}{h}(\frac{x-x-h}{x(x+h)})$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{1}{h}(\frac{-h}{x(x+h)})$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{-1}{x(x+h)}$

$f^{\prime }(x)=\frac{-1}{x^2}$

worked example: Differentiate $f(x)=\sqrt{x}$ using First Principles

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h}$

multiply by the conjugate

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})}$

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h}{h(\sqrt{x+h}+\sqrt{x})}$

$f^{\prime }(x)=\frac{1}{\sqrt{x+h}+\sqrt{x}}$

$f^{\prime }(x)=\frac{1}{2\sqrt{x}}$