Hi:) my teacher taught another method but she said we’ll only be learning it next year. I didn’t understand it , anyone able to explain? thank you so much! :) ♡

Differentiation can be used to find the gradient of the graph at a particular point.

To differentiate,

- bring down the power

- power minus 1

Differentiating y with respect to x term would be written as

Bringing down the power means to multiply the digit in the power to the equation first.

Let me show you an example.

Hence the gradient of the graph y=5x² at any point is given by the formula of 10x.

So at x=2,

gradient of graph is 10(2)= 20

Knowing the gradient, you can now find 2 points to plot your tangent using the gradient formula:

Let's look at example 3 again.

In this case, the graph is y=x³.

① Differentiate y with respect to x

② Find the value of the gradient at the point given.

When x=2,

This means that the gradient of the tangent at x=2 is 12.

So at x=2, y=8

Let y2 be 8 and x2 be 2.

Subst into the gradient formula:

Now you have an equation which shows how any coordinates that lie on the tangent are related to each other.

Use this to find 2 coordinates so you can draw a line through them to draw the tangent.

When x=3,

y= 12(3) -16

y= 36 -16

y= 20

when x= 1.5,

y= 12(1.5) -16

y= 2

Plot these 2 points: (3 ,20) and (1.5, 2)

Now draw the tangent through these 2 points.

Let's check:

Essentially, what we are trying to achieve is an accurate gradient value. However, we are not allowed to use differentiation ( Sec 4 A Math syllabus ) to answer the question. So use that as a side working to achieve an accurate tangent or you can also use that to check if your answer is close to the correct answer.

If the equation is y= x³-12 instead,

note that you have to differentiate x³ just like in example 3, ignoring the -12 since differentiating a constant would give you zero.

Further explanation:

Constants are basically _x⁰.

3= 3x⁰

4= 4x⁰

This is because x⁰= 1.

So if we differentiate 3,

we get d/dx (3x⁰)= 0(3x^-1)

This would give us 0 since anything multiplied by 0 is still 0.

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents