At this stage, you should be very familiar with gradients. You should know how to find the gradient of a straight line using rise over run or by inspecting the points on the line. The gradient measures steepness and, for a straight line, the gradient is the same at all points on the line. However, for a curve, such as a quadratic for example, the gradient is always changing. If you are given a function of the form $y=f(x)$, then the value of the gradient will depend on the value of $x$, i.e. the location of where you want to know the gradient. It follows that the gradient is also a function of $x$, we call it the DERIVATIVE and it is denoted as $\frac{dy}{dx}$ of $f$. See here to find out how to differentiate polynomials.
Example 1 – Find the gradient of the curve $y=x^2$ at the point (3,9).
We wish to find the gradient when x=3. The gradient is given by $\frac{dy}{dx}=2x$. Substituting x=3 into the gradient function tells us that the gradient at the given point is 6.
Example 2 – Find the gradient of the normal to the curve $y=3x^2-2x+7$ at the point (1,8).
The gradient of the curve is given by $\frac{dy}{dx}=6x-2$. At the point with x coordinate 1 the gradient is 4. Recall that if the gradient of the tangent to a curve (which is the same as the gradient of the curve at that point) is m, then the gradient of the normal to the curve at the point is -1/m. Hence, the gradient of the curve, $y=3x^2-2x+7$ at the point (1,8) is -1/4.