WORKED EXAMPLE

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.


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#Differentiation

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Calculate the derivative of the following function.

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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.

#2/ 3

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=limh0 fx+h-f(x)h

 

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: fx=x2.

 

First calculate f(x+h):

  •        we know that fx=x2    fx+h=x+h2
  •        expand x+h2    x2+2xh+h2

 

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

 

f'x=limh0fx+h-fxh=limh0x2+2xh+h2-x2h

 

Simplify: x2 and -x2 cancel each other, therefore then we can divide both the top and bottom by h:

 

f'x=limh0x2+2xh+h2-x2h=limh02xh+h2h=limh02x+h

 

Then since h heads towards 0, we are free to let h0:

 

f'x=limh02x+h=2x

 

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

Example #2

Find the derivatives of the following function from first principles: fx=x2+6x+9.

 

First calculate f(x+h):

  •   we know that fx=x2+6x+9    fx+h=x+h2+6x+h+9

 

  •   expand x+h2+6x+h+9    x2+2xh+h2+6x+6h+9

 

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

 

f'x=limh0fx+h-fxh=limh0x2+2xh+h2+6x+6h+9-x2-6x-9h

 

Simplify: x2 and -x2, 6x and -6x, 9 and -9 cancel each other, then we can divide both the top and bottom by h

 

limh02xh+h2+6hh=limh02x+h+6

 

Then since h heads towards 0, we are free to let h0:

 

limh02x+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 xn nxn-1
Multiplication by constant cf(x) cf'(x)
Sum rule fx+gx f'x+g'x

 

 

(1) derivative of xn :  fx=xn  f'x=nxn-1  
 

Example: f(x)=x4  f'x=4×x(4-1)=4x3

 

 

(2) If we differentiate kfx, where k is a constant, we get kf'x: kfx'=kf'(x)

 

Example: We have fx=4x6, to find f'x differentiate x6 then multiply by 4.

The derivative of x6: 6x5.

So f'x=4×6x5=24x5

  

 

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

 

Example: f(x)=x2+6x

To find f'x differentiate x2 and 6x separately and add up the results.

The derivative of x2=2x and the derivative of 6x=6, so: f'x=2x+6.

 

The derivatives of standard functions are in the formula booklet.

Example #1

Find the derivative of the following function: fx=2x3+45x2+6x+3

 

Differentiate each term separately then and add up the results.

 

f'x=2×3x2+45×2x+6+0=6x2+85x+6

 

Example #2

Find the derivative of the following function: gx=1x3.

 

First use the laws of exponents to rewrite the function in the form xn

1x3=1x13=x-13

 

Then use the differentiation formula:

 

g'x=-13x(-13)-1=-13x-43

Example #3

Find the derivative of the following function: hx=4ex+tanx.

 

Differentiate each term separately then and add up the results.

 

h'x=4ex+1cos2x

 

Example #4

Find the derivative of the following function: ix=5sinx+4cosx.

 

Differentiate each term separately then and add up the results.

 

i'x=5cosx-4sinx