Gradients & Derivatives

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.