guide to differentiation - start application questions

differentiation is trying to find the derivative of an expression, equation or function

in differential calculus, the letter $d$ when prepended to a variable means “a little bit of” the variable

so $dx$ just means a infinitely small bit of $x$, in other words an element of $x$ or the differential of $x$

the word “integral” simply means “the whole”
if it is the integral from 2 to 3 of $dx$, that is all the little bits of $dx$ in that region together

suppose we have two variables that depend on one another
an alteration in one will cause an alteration in the other
let us call the variables $x$ and $y$

suppose we change $x$, to alter it or imagine it to be altered, by adding a bit we call $dx$
we are causing $x$ to become $x+dx$
because $x$ has been altered, $y$ will have altered also, and it will have become $y+dy$

$dx$ and $dy$ are not variables or constants.
their sole purpose is to act as a placeholder for the concept of change, used to compare to each other

in differential calculus we search for the ratio $\frac{dy}{dx}$ which is a ratio of how the changes are correlated - this is called differentiating
for no reason, we call the ratio $\frac{dy}{dx}$ the differential coefficient of $y$ with respect to $x$

for example, let $x$ and $y$ be respectively the base and height of a right angled triangle of which the slope is 30 degrees

because the angle is 30 degrees, $y=x\tan (30°)$
if we change $x$ to become $x+dx$, then $y$ would then become $y+dy$
$y+dy=(x+dx)\cdot \tan (30°)$
when $dy$ is 1, $dx$ must be $1.73$
expressed as a ratio, $\frac{dy}{dx}$ is $\frac{1}{1.73}$
we just did differentiation! hooray!

a positive $dx$ can correspond to a negative $dy$
for example, suppose we have a ladder against a wall and the angle is unknown

suppose when x is 19 m, y is 180 m
when $dx$ is 1, what will $dy$ be?
we know that the length of the ladder is constant, $\sqrt{180^2+19^2}=181$ m

because $x+dx=20$, we know that $\sqrt{20^2+(y+dy{)}^2}=181$
$400+(y+dy{)}^2=32761$
$y+d{y}^2=32361$
$y+dy=\pm \sqrt{32361}$
we know that a length has to be positive
$y+dy=\sqrt{32361}$
we know that y is 180
$dy=-0.11$

$\frac{dy}{dx}=\frac{-0.11}{1}$

what is meaning of implicit function and explicit function?
explicit functions express $x$ in terms of $y$ or express $y$ in terms of $x$
for example, $y^2+x^2=1$ is implicit whereas $y=\sqrt{1-x^2}$ is explicit

whenever we use differentials $dx$, $dy$, etc, the existence of a correlation between the variables is shown, expressed with functions

when the exact function between several quantities x, y, z is unknown or is not convenient to state it, it is assumed that there is some sort of function between these variables and one cannot alter any of them without affecting the other ones
this function is expressed implicitly with $F(x,y,z)$ or explicitly with $x=F(y,z)\vee y=F(x,z)\vee z=F(x,y)$

suppose we have the equation $y=x^2$
let us change x so that it becomes $x+dx$
this causes a change in y so now y becomes $y+dy$
now begins the differentiation, aka finding $\frac{dy}{dx}$
$y+dy=(x+dx{)}^2$
$y+dy=x^2+2x\cdot dx+(dx{)}^2$
so what does $(dx{)}^2$ mean? it is a infinitely tiny amount squared which places it in the second order of smallness, nonexistent compared to the other terms
$y+dy=x^2+2x\cdot dx$
substitute $y=x^2$
$dy=2x\cdot dx$
$\frac{dy}{dx}=2x$

suppose we have the equation $y=x^3$
we let $x$ become $x+dx$ and $y$ will become $y+dy$
$y+dy=(x+dx{)}^3$
$y+dy=x^3+3x^2\cdot dx+3x(dx{)}^2+(dx{)}^3$
remove the too small things
$y+dy=x^3+3x^2\cdot dx$
substitute $y=x^3$
$\frac{dy}{dx}=3x^2$

have a look at representing functions
keep in mind that when we differentiate a function such as $y=x^2$, we are saying that $f(x)=x^2$ and the $\frac{dy}{dx}$ can be simply represented as $\frac{d}{dx}[x^2]$ and $f^{\prime }$ where $f(x)=x^2$

in other words, the following statements represent the exact same thing

• $\frac{dy}{dx}$ for $y=x^2$
• $\frac{d}{dx}[x^2]$
• $f^{\prime }(x)$ for $f(x)=x^2$

now you are ready to learn the power rule
if $f(x)=x^n$ and $n$ is a positive integer, then $f^{\prime }(x)=n\cdot x^{n-1}$

for example, if $f(x)=x^7$, $f^{\prime }(x)=7x^6$

worked example: $f(x)=5x^3-10x^2+7$
$f^{\prime }(x)=5(3x^2)-10(2x)+0$
$f^{\prime }(x)=15x^2-20x$

if $f(x)=g(x)-h(x)$ then $f\prime (x)=g\prime (x)-h\prime (x)$

see derivatives page and you can see the differential rules

we can practice the power rule by solving these

note: questions that require you to find the derivative of an expression uses one of these formats. just wanted to say that these are different ways to say the same thing

• find the derivative with respect to x ($\frac{d}{dx}$) of this expression
• for $y=...$, find $\frac{dy}{dx}$
• differentiate the equation $y=...$
• for $f(x)=...$, find $f^{\prime }(x)$
• find the equation of the tangent line to the graph of $f(x)$

worked example: find derivative of each of these expressions

$\frac{d}{dx}{x}^2+4x=2x+4$
$\frac{d}{dx}-x^3+2x^2=-3x^2+4x$
$\frac{d}{dx}-10x^3-2x^2+2=-30x^2-4x$