# 5.1 Differentiation (first principles, rules) and sketching graphs  (Page 2/3)

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## Summary of differentiation rules

Given two functions, $f\left(x\right)$ and $g\left(X\right)$ we know that:

 $\frac{d}{dx}b=0$ $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$ $\frac{d}{dx}\left(kf\right)=k\frac{df}{dx}$ $\frac{d}{dx}\left(f+g\right)=\frac{df}{dx}+\frac{dg}{dx}$

## Rules of differentiation

1. Find ${f}^{\text{'}}\left(x\right)$ if $f\left(x\right)=\frac{{x}^{2}-5x+6}{x-2}$ .
2. Find ${f}^{\text{'}}\left(y\right)$ if $f\left(y\right)=\sqrt{y}$ .
3. Find ${f}^{\text{'}}\left(z\right)$ if $f\left(z\right)=\left(z-1\right)\left(z+1\right)$ .
4. Determine $\frac{dy}{dx}$ if $y=\frac{{x}^{3}+2\sqrt{x}-3}{x}$ .
5. Determine the derivative of $y=\sqrt{{x}^{3}}+\frac{1}{3{x}^{3}}$ .

## Applying differentiation to draw graphs

Thus far we have learnt about how to differentiate various functions, but I am sure that you are beginning to ask, What is the point of learning about derivatives? Well, we know one important fact about a derivative: it is a gradient. So, any problems involving the calculations of gradients or rates of change can use derivatives. One simple application is to draw graphs of functions by firstly determine the gradients of straight lines and secondly to determine the turning points of the graph.

## Finding equations of tangents to curves

In "Average Gradient and Gradient at a Point" we saw that finding the gradient of a tangent to a curve is the same as finding the gradient (or slope) of the same curve at the point of the tangent. We also saw that the gradient of a function at a point is just its derivative.

Since we have the gradient of the tangent and the point on the curve through which the tangent passes, we can find the equation of the tangent.

Find the equation of the tangent to the curve $y={x}^{2}$ at the point (1,1) and draw both functions.

1. We are required to determine the equation of the tangent to the curve defined by $y={x}^{2}$ at the point (1,1). The tangent is a straight line and we can find the equation by using derivatives to find the gradient of the straight line. Then we will have the gradient and one point on the line, so we can find the equation using: $y-{y}_{1}=m\left(x-{x}_{1}\right)$ from grade 11 Coordinate Geometry.

2. Using our rules of differentiation we get: ${y}^{\text{'}}=2x$

3. In order to determine the gradient at the point (1,1), we substitute the $x$ -value into the equation for the derivative. So, ${y}^{\text{'}}$ at $x=1$ is: $m=2\left(1\right)=2$

4. $\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-1& =& \left(2\right)\left(x-1\right)\hfill \\ \hfill y& =& 2x-2+1\hfill \\ \hfill y& =& 2x-1\hfill \end{array}$
5. The equation of the tangent to the curve defined by $y={x}^{2}$ at the point (1,1) is $y=2x-1$ .

## Curve sketching

Differentiation can be used to sketch the graphs of functions, by helping determine the turning points. We know that if a graph is increasing on an interval and reaches a turning point, then the graph will start decreasing after the turning point. The turning point is also known as a stationary point because the gradient at a turning point is 0. We can then use this information to calculate turning points, by calculating the points at which the derivative of a function is 0.

If $x=a$ is a turning point of $f\left(x\right)$ , then: ${f}^{\text{'}}\left(a\right)=0$ This means that the derivative is 0 at a turning point.

Take the graph of $y={x}^{2}$ as an example. We know that the graph of this function has a turning point at (0,0), but we can use the derivative of the function: ${y}^{\text{'}}=2x$ and set it equal to 0 to find the $x$ -value for which the graph has a turning point.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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