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

This simplifies to

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 $2\times 3$ , that is 6, and when $x=1$ the gradient is $2\times 1$ , 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 $y^{\prime }$ .

Exercises

1. Carry out the procedure above for the function $y=3x^2$ :
   1. Let $A$ be the point $\left(a,3a^2\right)$ .
   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 $A$ .

Answer