## BASICS

### OF DERIVATIVES

#Differentiation

With differentiation we jump into calculus. This onepager first presents the “classical” way of differentiation, then shows how to work with the rules.

Explanations
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## PRACTICE [TASKS ACCORDING TO THIS SUBTOPIC]

#Differentiation

### TASK

Calculate the derivative of the following function.

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Other Onepagers in #Differentiation
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##### #1/ 3

Basic definitions

A tangent to a curve is a straight line that only touches the curve at a single point and does not go through it.

The gradient of a function at a certain point is the gradient of the tangent at that point.

The derivative of a function is also a function which gives the gradient of f(x) at all points of its curve.

Differentiation is the process of finding the derivative of a function.

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First principles

Since the derivative is the rate of change in the value of the function (i.e. in X) to change in the independent variable (i.e. in Y), we can use the following formula to calculate the derivative:

f’(x) is the derivative of f(x).

We can use this definition to find the derivative of simple polynomial functions. To understand how to use the formula, take a look at the worked examples.

The first principles is in the formula booklet.

#### Example #1

Find the derivatives of the following function from first principles: $f\left(x\right)={x}^{2}$.

First calculate f(x+h):

•        we know that
•        expand

Substitute f(x+h) and f(x) into the formula:

$f\text{'}\left(x\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\mathrm{lim}}\frac{{x}^{2}+2xh+{h}^{2}-{x}^{2}}{h}$

Simplify: ${x}^{2}$ and $-{x}^{2}$ cancel each other, therefore then we can divide both the top and bottom by h:

$f\text{'}\left(x\right)=\underset{h\to 0}{\mathrm{lim}}\frac{{x}^{2}+2xh+{h}^{2}-{x}^{2}}{h}=\underset{h\to 0}{\mathrm{lim}}\frac{2xh+{h}^{2}}{h}=\underset{h\to 0}{\mathrm{lim}}2x+h$

Then since h heads towards 0, we are free to let $h\to 0$:

$f\text{'}\left(x\right)=\underset{h\to 0}{\mathrm{lim}}2x+h=2x$

So the derivative of f(x) is  f’(x)=2x.

#### Example #2

Find the derivatives of the following function from first principles: $f\left(x\right)={x}^{2}+6x+9$.

First calculate f(x+h):

•   we know that

•   expand

Substitute f(x+h) and f(x) into the formula:

$f\text{'}\left(x\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\mathrm{lim}}\frac{{x}^{2}+2xh+{h}^{2}+6x+6h+9-{x}^{2}-6x-9}{h}$

Simplify: ${x}^{2}$ and $-{x}^{2}$, $6x$ and $-6x$, $9$ and $-9$ cancel each other, then we can divide both the top and bottom by h

$\underset{h\to 0}{\mathrm{lim}}\frac{2xh+{h}^{2}+6h}{h}=\underset{h\to 0}{\mathrm{lim}}2x+h+6$

Then since h heads towards 0, we are free to let $h\to 0$:

$\underset{h\to 0}{\mathrm{lim}}2x+h+6=2x+6$

So the derivative of f(x) is  f’(x)=2x+6.

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Rules of differentiation

In practice the usual way to find derivatives is to use the rules.

 Rules Function Derivative Power rule ${x}^{n}$ $n{x}^{n-1}$ Multiplication by constant $cf\left(x\right)$ $cf\text{'}\left(x\right)$ Sum rule $f\left(x\right)+g\left(x\right)$ $f\text{'}\left(x\right)+g\text{'}\left(x\right)$

(1) derivative of ${x}^{n}$ :

Example:

(2) If we differentiate $kf\left(x\right)$, where $k$ is a constant, we get $kf\text{'}\left(x\right)$: $\left(kf\left(x\right)\right)\text{'}=kf\text{'}\left(x\right)$

Example: We have $f\left(x\right)=4{x}^{6}$, to find $f\text{'}\left(x\right)$ differentiate ${x}^{6}$ then multiply by $4$.

The derivative of ${x}^{6}$: $6{x}^{5}$.

So $f\text{'}\left(x\right)=4×6{x}^{5}=24{x}^{5}$

(3) To differentiate a sum, we can differentiate its terms one at a time and then add up the results: $\left(f\left(x\right)+g\left(x\right)\right)\text{'}=f\text{'}\left(x\right)+g\text{'}\left(x\right)$

Example: $f\left(x\right)={x}^{2}+6x$

To find $f\text{'}\left(x\right)$ differentiate ${x}^{2}$ and $6x$ separately and add up the results.

The derivative of ${x}^{2}=2x$ and the derivative of $6x=6$, so: $f\text{'}\left(x\right)=2x+6$.

The derivatives of standard functions are in the formula booklet.

#### Example #1

Find the derivative of the following function: $f\left(x\right)=2{x}^{3}+\frac{4}{5}{x}^{2}+6x+3$

Differentiate each term separately then and add up the results.

$f\text{'}\left(x\right)=2×3{x}^{2}+\frac{4}{5}×2x+6+0=6{x}^{2}+\frac{8}{5}x+6$

#### Example #2

Find the derivative of the following function: $g\left(x\right)=\frac{1}{\sqrt[3]{x}}$.

First use the laws of exponents to rewrite the function in the form ${x}^{n}$

$\frac{1}{\sqrt[3]{x}}=\frac{1}{{x}^{1}{3}}}={x}^{-1}{3}}$

Then use the differentiation formula:

$g\text{'}\left(x\right)=\frac{-1}{3}{x}^{\left(-1}{3}\right)-1}=\frac{-1}{3}{x}^{-4}{3}}$

#### Example #3

Find the derivative of the following function: $h\left(x\right)=4{e}^{x}+\mathrm{tan}\left(x\right)$.

Differentiate each term separately then and add up the results.

$h\text{'}\left(x\right)=4{e}^{x}+\frac{1}{{\mathrm{cos}}^{2}x}$

#### Example #4

Find the derivative of the following function: $i\left(x\right)=5\mathrm{sin}\left(x\right)+4\mathrm{cos}\left(x\right)$.

Differentiate each term separately then and add up the results.

$i\text{'}\left(x\right)=5\mathrm{cos}\left(x\right)-4\mathrm{sin}\left(x\right)$