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 , then the value of the gradient will depend on the value of , i.e. the location of where you want to know the gradient. It follows that the gradient is also a function of , we call it the DERIVATIVE and it is denoted as of . See here to find out how to differentiate polynomials.
Example 1 – Find the gradient of the curve at the point (3,9).
We wish to find the gradient when x=3. The gradient is given by . 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 at the point (1,8).
The gradient of the curve is given by . 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, 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 . 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 . Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.
Example 1– Find the range of values of x for which the graph of has a positive gradient.
Differentiating y gives . This is positive when , i.e. when 2x is greater than 5. This gives the solution .
Example 2 – Explain why the gradient of 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.
which is 3 lots of a square number. Irrespective of the value of x that is put in, this number will always be positive.