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.

## Increasing & Decreasing Functions

Recall that upward sloping straight lines have a positive gradient whereas downward sloping straight lines have a negative gradient. The same applies to curves. Gradients on a curve are always changing but an upward sloping curve has a positive gradient and a downward sloping curve has a negative gradient.

Recall the graph of $y=x^2$. You will notice that for positive x, the graph has a positive gradient; for negative x the graph has a negative gradient; and for x=0 the gradient is also 0.

This can be seen from the gradient function $\frac{dy}{dx}=2x$. Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.

Example 1Find the range of values of x for which the graph of $y=x^2-5x+4$ has a positive gradient.

Differentiating y gives $\frac{dy}{dx}=2x-5$. This is positive when $2x-5\textgreater 0$, i.e. when 2x is greater than 5. This gives the solution $x\textgreater 2.5$.

Example 2 Explain why the gradient of $y=x^3$ is never negative.

You can see from the graph of x cubed that it never has a negative gradient but we show it using differentiation.

$\frac{dy}{dx}=3x^2$ which is 3 lots of a square number. Irrespective of the value of x that is put in, this number will always be positive.