### 3 Finding the gradient at a general point

We now carry out the previous procedure more mathematically. Consider the graph of $y\left(x\right)={x}^{2}$ in Figure 6. Let point $A$ be any point with coordinates $\left(a,{a}^{2}\right)$ , and let point $B$ be a second point with $x$ coordinate $\left(a+h\right)$ .

The $y$ coordinate at $A$ is ${a}^{2}$ , because $A$ lies on the graph $y={x}^{2}$ .

Similarly the $y$ coordinate at $B$ is ${\left(a+h\right)}^{2}$ .

Therefore the gradient of the chord $AB$ is

$\frac{{\left(a+h\right)}^{2}-{a}^{2}}{h}$

This simplifies to

$\frac{{a}^{2}+2ha+{h}^{2}-{a}^{2}}{h}=\frac{2ha+{h}^{2}}{h}=\frac{h\left(2a+h\right)}{h}=2a+h$
This is the gradient of the line $AB$ . As we let $B$ move closer to $A$ the value of $h$ gets smaller and smaller and eventually tends to zero. We write this as $h\to 0$ .

Now, as $h\to 0$ , the gradient of $AB$ tends to $2a$ . Thus the gradient of the tangent to the curve at point $A$ is $2a$ . Because $A$ is an arbitrary point, this result gives us a formula for finding the gradient of the graph of $y={x}^{2}$ at any point: the gradient is simply twice the $x$ coordinate there . For example when $x=3$ the gradient is $2×3$ , that is 6, and when $x=1$ the gradient is $2×1$ , that is 2 as we saw in the previous subsection.

Figure 6

Generally, at a point whose coordinate is $x$ the gradient is given by $2x$ . The function, $2x$ which gives the gradient of $y={x}^{2}$ is called the derivative of $y$ with respect to $x$ . It has other names too including the rate of change of $y$ with respect to $x$ .

A special notation is used to represent the derivative. It is not a particularly user-friendly notation but it is important to get used to it anyway. We write the derivative as $\frac{dy}{dx}$ , pronounced ‘dee $y$ over dee $x$ ’ or ‘dee $y$ by dee $x$ ’ or even ‘dee $y$ , dee $x$ ’.

$\frac{dy}{dx}$ is not a fraction - so you can’t do things like cancel the $d$ ’s - just remember that it is the symbol or notation for the derivative. An alternative notation for the derivative is ${y}^{\prime }$ .

##### Key Point 2

The derivative of  $y\left(x\right)$  is written  $\frac{dy}{dx}$  or  ${y}^{\prime }\left(x\right)$ or simply  ${y}^{\prime }$

##### Exercises
1. Carry out the procedure above for the function $y=3{x}^{2}$ :
1. Let $A$ be the point $\left(a,3{a}^{2}\right)$ .
2. Let $B$ be the point $\left(a+h,3{\left(a+h\right)}^{2}\right)$ .
3. Find the gradient of the line $AB$ .
4. Let $h\to 0$ to find the gradient of the curve at $A$ .
2. Carry out the procedure above for the function $y={x}^{3}$ :
1. Let $A$ be the point $\left(a,{a}^{3}\right)$ .
2. Let $B$ be the point $\left(a+h,{\left(a+h\right)}^{3}\right)$ .
3. Find the gradient of the line $AB$ .
4. Let $h\to 0$ to find the gradient of the curve at $A$ .

1. gradient $AB=6a+3h$ , gradient at $A=6a$ . So, if $y=3{x}^{2}$ , $\frac{dy}{dx}=6x$ ,

2. gradient $AB=3{a}^{2}+3ah+{h}^{2}$ , gradient at $A=3{a}^{2}$ . So, if $y={x}^{3}$ , $\frac{dy}{dx}=3{x}^{2}$ .