Gradients & Derivatives

GradientsDerivativesAt 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.

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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.