# 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.

#### Questions & Answers

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Renato
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?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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