Differentiation of Polynomials

Before You Watch

This video continues directly on from Rates of Change and Differentiation, in which the concept of a rate of change was introduced and we investigated the need to know the value of instantaneous rates of change. The term ‘to differentiate’ was also discussed. Differentiation is the process of calculating a rate of change. This video describes how to differentiate a specific class of equations known as polynomials. So make sure you’ve seen Rates of Change and Differentiation recently before watching this video.

This video also builds upon earlier algebraic concepts such as indices, including negative indices, and linear equations. It is important to be comfortable with algebra and manipulating algebraic equations before continuing with calculus, so watch those videos again if you need to, then come back.

The Video

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Some Practice Questions

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Differentiation of Polynomials

Differentiation of Polynomials

For this quiz, you will be required to differentiate each of the expressions. You will then have to enter answers that have indices. To do this, use the symbol ^ and order the terms from highest power to lowest. For example, if the answer to a question was , you would type into the answer box:

-2p^3+6p+7.

Differentiate the following:

$y=x^2$

$\frac{dy}{dx}=$

Incorrect!

Correct!

Answer:

$\begin{align*} y&=x^2\\ \frac{dy}{dx}&=2\times x^{(2-1)} \\ \frac{dy}{dx}&=2x \\ \mbox{Answer: }\frac{dy}{dx}&=2x \end{align*}$

$p=4t^2-3t$

$\frac{dp}{dt}=$

Incorrect!

Correct!

Answer:

$\begin{align*} p&=4t^2-3t\\ \frac{dp}{dt}&=2\times 4t^{(2-1)}-1\times 3t^{(1-1)} \\ &=8t^1-3t^{0} \\ \frac{dp}{dt}&=8t-3t \\ \mbox{Answer: }\frac{dp}{dt}&=8t-3t \end{align*}$

$j=5r^6-7r^4+5r^2$

$\frac{dj}{dr}=$

Incorrect!

Correct!

Answer:

$\begin{align*} j&=5r^6-7r^4+5r^2\\ \frac{dj}{dr}&=6\times 5r^{(6-1)}-4\times 7r^{(4-1)}+2\times 5r^{(2-1)} \\ \frac{dj}{dr}&=30r^{5}-28r^{3}+10r \\ \mbox{Answer: }\frac{dj}{dr}&=30r^{5}-28r^{3}+10r \end{align*}$

$q=(7x+3x^2)^2$

$\frac{dq}{dx}=$

Incorrect!

Correct!

Hint: expand out the brackets first.

Answer:

$\begin{align*} q&=(7x+3x^2)^2\\ &=(7x+3x^2)\times (7x+3x^2) \\ &=7x\times 7x +7x \times 3x^2 + 3x^2\times 7x + 3x^2 \times 3x^2 \\ &=49x^2 +21x^3 + 21x^3 + 9x^4 \\ q&=9x^4 + 42x^3 + 49x^2 \\ \frac{dq}{dx}&= 4\times 9x^{(4-1)} + 3\times 42x^{(3-1)} + 2\times 49x^{(2-1)} \\ \frac{dq}{dx}&= 36x^3 + 126x^2 + 98x \\ \mbox{Answer: }\frac{dq}{dx}&= 36x^3 + 126x^2 + 98x \end{align*}$

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Now What?

By now you will be familiar with the three videos that introduce the differential branch of calculus: Introduction to Calculus, Rates of Change and Differentiation and this topic, Differentiation of Polynomials. You should understand the core concepts of calculus and know what a rate of change is. You will also know that differentiation is all about calculating the rate of change, and know how to differentiate one category of equations, the polynomials. From here there are two main directions you can go.

One option is to explore how to differentiate other types of equations, such as those involving trigonometry, or exponentials. To do this you should consider looking at sites such as the Khan Academy: https://www.khanacademy.org/math/differential-calculus/taking-derivatives

Alternatively, you could investigate the other branch of calculus, integral calculus. This is introduced in the video Integration.

But When I am Going to Use This

Calculus is the mathematical study of how things change relative to one another. For instance, velocity (or speed) is a change of position over a change in time, and acceleration is a change in velocity over a change in time – so any motion is studied using calculus. Other examples include the flow of water through pipes over time, or changing commodity prices against demand. Because change is everywhere, the potential applications for calculus are endless, particularly in engineering and science. Calculus is necessary knowledge for any degree related to engineering or science.

Differentiation of Polynomials

Before You Watch

Before You Watch

The Video

The Video

Some Practice Questions

Some Practice Questions

Differentiation of Polynomials

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Answer:

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Now What?

But When I am Going to Use This

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