0580 · 2.12Differentiation

Gradients and tangents

0:00A curve can climb, flatten out, and then fall. Its height tells us where it is, but its gradient tells us how it is changing. Differentiation connects those two ideas. By the end of this video, you will find gradients and explain why a turning point is a maximum or minimum. For Cambridge International Mathematics, this topic belongs to the Extended content. We will work with powers of x whose exponents are positive integers or zero.

0:24The functions can contain up to three terms, with rational coefficients. That includes whole numbers, fractions and terminating decimals multiplying those powers. You need to remember how to calculate the gradient of a straight line. Gradient equals the change in y divided by the change in x. You also need to substitute into expressions and solve simple equations. We will use those skills without reteaching them in detail. For a straight line, the gradient is constant wherever you measure it.

0:52Here, moving three units to the right takes us six units upwards. Dividing six by three gives a gradient of two. Every section of this line has the same steepness, so one gradient describes the entire line. A curve behaves differently because its steepness can change from point to point. Joining two points on a curve gives the gradient between those points. That does not usually give the gradient at one particular point.

1:17To measure the steepness at a point, we need a tangent. A tangent is a straight line with the same direction as the curve at the chosen point. Imagine zooming in until the curve looks almost straight around that point. The tangent follows that local direction. We define the gradient of the curve there as the gradient of its tangent. On a printed graph, draw a tangent with a ruler at the required point.

1:39Adjust its direction so that it follows the curve closely nearby. Then choose two widely separated points on your tangent. A large gradient triangle makes small drawing and reading errors less important. Our first exam-style example asks us to estimate the gradient at the point with x equal to two on this graph. Draw the tangent through the marked point. From that tangent, read the points with coordinates one comma zero and three comma eight.

2:04These are points on the tangent, not both on the curve. The change in y is 8 − 0, which gives eight. For the horizontal change, 3 − 1 gives two. Dividing eight by two gives an estimated gradient of four. We call this an estimate because drawing the tangent and reading the graph introduce uncertainty. Keep the coordinate order consistent when subtracting. If you move from the first point to the second, use that same movement for both coordinate changes.

2:33Never select two convenient points on the curve instead. Their connecting line would generally measure a different gradient from the tangent you need.

Derivatives from small changes

2:40Drawing tangents works, but an equation lets us calculate gradients directly. Differentiation produces a new expression that gives the curve’s gradient at any chosen x value. This expression is called the derivative. It describes steepness rather than height, even though both expressions use the same variable. The curve y = x² helps explain where a derivative comes from. Start at any x value, then move a small distance called h to the right.

3:07The new input is x + h. Its height is the square of x + h, rather than x². Expanding the new height gives x squared plus two times x times h plus h squared. Subtracting the original height leaves two times x times h plus h squared. Dividing by the horizontal change h gives 2x + h. That is the gradient between the two nearby points. As the gap h becomes smaller, this gradient gets closer to 2x.

3:36The line through the nearby points approaches the tangent’s direction. So the derivative of x² is 2x. You do not need to reproduce this reasoning in an examination, but it explains what differentiation calculates.

Power rule and gradients

3:49For the powers in this topic, differentiation follows a power rule. Multiply the coefficient by the original exponent, then reduce the exponent by one. For a times x to the power n, the derivative is a × n × x to the power n − 1, where the whole of n − 1 is the new power. Apply this rule when n is a positive integer. For example, differentiating x³ gives 3x².

4:13A term of 5x to the power four becomes 20x³. The original exponent supplies the multiplier, while the remaining power drops by one. Our small-gap reasoning for x² shows the underlying idea behind this pattern. A linear term follows the same rule, so 7x differentiates to seven. Its graph is a straight line with gradient seven everywhere. A constant such as five differentiates to zero because its graph is horizontal.

4:39This also covers the zero exponent, since x to the power zero gives a constant term. Differentiate a sum one term at a time, keeping the signs attached to their terms. A rational coefficient stays involved in the multiplication. The notation shown on screen is read as dee y by dee x. It represents the derivative of y with respect to x, giving the gradient of the original curve. Our second exam-style example gives y equal to one half of x to the power four, minus 3x², plus five.

5:09Find the gradient when x = 2. Differentiating the first term gives one half of 4x³. The second term becomes −3 × 2x, and the constant becomes zero. Simplifying gives dee y by dee x equal to 2x³ − 6x. Substitute two into this derivative to find the gradient. 2³ is eight, so the expression becomes two multiplied by eight minus twelve. This gives 16 − 12, leaving a gradient of four.

5:38Substitution belongs after differentiation when you are finding a gradient. Substituting into the original equation would give the curve’s height instead. Also, the derivative is an expression until an x value is supplied. Different positions can produce different gradients from that same expression.

Finding stationary points

5:55A stationary point is a point where the gradient is zero. Its tangent is horizontal, even if the point is high above or below the x axis. At a maximum turning point, the curve changes from rising to falling. A minimum turning point is the reverse, going from falling to rising. The word local means we are comparing the point with nearby parts of the curve. A local maximum need not be the highest point anywhere on the entire graph.

6:20To locate stationary points, set the derivative equal to zero and solve. Then substitute each solution into the original equation to find its y coordinate. Our third example asks for the stationary points of y equal to x³ − 3x² + 2. Differentiating gives 3x² − 6x. Set this derivative equal to zero because a stationary point has zero gradient. Factorising gives 3x multiplied by the expression x − 2 equal to zero.

6:49A product is zero when at least one of its factors is zero. Therefore, x = 0 or x = 2. Substituting zero into the original equation gives y equal to two. At x equal to two, 8 − 12 + 2 gives negative two. The stationary points therefore have coordinates zero comma two and two comma negative two. Finding these coordinates does not yet establish their types. We must show how the curve behaves around them.

7:15Inspecting the gradient either side provides a direct way to discriminate between maxima and minima. To test the gradient before the first stationary point, substitute negative one into the derivative. This gives three multiplied by one plus six, which equals nine. Between the stationary points, using x equal to one gives 3 − 6, or negative three. After the second point, x equal to three gives 27 − 18, or nine.

7:41At x equal to zero, the gradient changes from positive to negative. The curve rises and then falls, so the point with coordinates zero comma two is a maximum. The opposite happens at x equal to two, where a falling curve starts to rise. That makes two comma negative two a minimum. Another accepted method uses the second derivative, also called the second differential. Differentiate the first derivative again to obtain it.

8:07Here, 3x² − 6x becomes 6x − 6. The second derivative tells us how the gradient itself is changing. At a stationary point, a negative second derivative identifies a maximum. A positive value identifies a minimum. In our example, substituting zero gives negative six, while substituting two gives positive six. If the second derivative is zero, that test alone does not decide the type, so inspect the gradient either side.

8:35This method makes sense because decreasing gradients produce a downward bend around a maximum. Around a minimum, the gradients increase instead, giving an upward bend. An accurate sketch can also identify the type when it clearly shows the surrounding shape. Do not decide from the point’s height or the sign of its y coordinate.

Stationary point exam practice

8:54A final exam-style question asks you to find and identify the stationary point of y equal to −2x² + 12x + 1. Differentiation gives −4x + 12. Setting this equal to zero gives −4x equal to negative twelve. Dividing by negative four gives x equal to three. Substitute three into the original equation to obtain negative two multiplied by three squared plus twelve multiplied by three plus one. This simplifies to −18 + 36 + 1, giving nineteen.

9:27Differentiating again gives a second derivative of negative four. Since this is negative, the stationary point with coordinates three comma nineteen is a maximum. Common mistakes include keeping a constant unchanged and forgetting to reduce a power. A constant contributes no gradient, so its derivative must be zero. Another mistake is setting y equal to zero when seeking stationary points. That finds x axis intersections instead of points with horizontal tangents.

9:54These differentiation and algebra methods must work without a calculator on Paper Two. Paper Four allows a calculator, but clear working remains important. Keep exact values where possible and do not round during a calculation. When a numerical answer needs rounding, use three significant figures unless the question specifies otherwise. Differentiation turns a height equation into a gradient equation. A tangent estimates that gradient graphically, while the derivative calculates it from the function.

10:23Stationary points occur where the derivative is zero. Their coordinates come from the original equation, and their types follow from a sketch, gradient signs or the second derivative.