The derivative of a polynomial · Years 11–12
In The derivative of a polynomial (Years 11–12) you practise 9 skills: Average rate of change · The derivative at a point · The rules of differentiation · The tangent to a graph · Turning points · Where a function rises and falls · Intercepts and sign · The graph and the graph of its derivative · Finding a parameter. The average rate of change of a function between two points is the gradient of the chord joining them. As the second point comes closer to the first, the chords close in on the tangent, and their gradients close in on the derivative at the point. The derivative of x to the power of n is n times x to the power of n minus one, the derivative of a constant is zero, and each term is differentiated on its own.
The tangent at a point: its gradient is the derivative there, and with the point itself you get the equation of the line. Where the derivative is positive the function rises, where it is negative the function falls. At a turning point the derivative is zero and changes sign: from positive to negative it is a maximum, from negative to positive a minimum. To investigate a function, find its domain, where it meets the axes, its turning points and where it rises and falls. On a closed interval, the largest value is at a turning point or at one of the ends.
Tip: When the derivative is zero, check its sign on both sides. A zero alone does not mean there is a turning point.
What you learn
Average rate of change
For example: Here is the function f(x) = −4x + 6 . Find its average rate of change on the interval −2 ≤ x ≤ 0 .
Answer and solution
The answer: −4
f(0) = 6 f(−2) = 14 (6 − 14) ÷ (0 − (−2)) = −8 ÷ 2 = −4 A linear function has the same rate on every interval: its gradient.
The derivative at a point
For example: Here is the function f(x) = −3x² + 6x − 5 . Find f′(2) .
Answer and solution
The answer: −6
Differentiate each term: f′(x) = −6x + 6 f′(2) = −6 × 2 + 6 = −6
The rules of differentiation
For example: Here is the function f(x) = −4x² + 8x − 7 . Find its derivative.
Answer and solution
The answer: −8x + 8
Differentiate each term on its own: f′(x) = −8x + 8
The tangent to a graph
For example: Find the gradient of the tangent to the graph of the function f(x) = −3x² + 6x − 5 at the point where x = 0 .
Answer and solution
The answer: 6
f′(x) = −6x + 6 m = f′(0) = −6 × 0 + 6 = 6
Turning points
For example: Find the turning point of the function f(x) = x² − 6x + 4 .
Answer and solution
The answer: (3, −5)
f′(x) = 2x − 6 = 0 x = 3 For x < 3 the derivative is negative, and for x > 3 it is positive. It is a minimum point: f(3) = 3² − 6 × 3 + 4 = −5 (3, −5)
Where a function rises and falls
For example: Where does the function f(x) = −2x² − 12x increase ?
- x > −3
- Every number
- x > 3
- x < −3
Answer and solution
The answer: x < −3
f′(x) = −4x − 12 = 0 x = −3 For x < −3 the derivative is positive, and for x > −3 it is negative. So the function increases for x < −3 .
Intercepts and sign
For example: Here is the function f(x) = (x + 3)²(x − 4) . Find its zeros, the values of x where f(x) = 0 .
Answer and solution
The answer: x = −3 or x = 4
x + 3 = 0 or x − 4 = 0 . So x = −3 or x = 4 . At x = −3 the factor is squared, so the graph only touches the axis there and does not cross it.
The graph and the graph of its derivative
For example: The first picture shows the graph of the function y = f(x) . Which of the graphs after it is the graph of its derivative?
Graph A: Graph B:- Graph A
- Graph B
Answer and solution
The answer: Graph A
The parabola falls until its vertex and rises after it. So the derivative is negative before the vertex, 0 at the vertex and positive after it: a rising line that crosses the horizontal axis at x = 2 . That is Graph A .
Finding a parameter
For example: Here is the function f(x) = −3x² + kx . You also know that f′(3) = −23 . Find k .
Answer and solution
The answer: −5
f′(x) = −6x + k f′(3) = −6 × 3 + k = −23 −18 + k = −23 k = −5
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4) and selected AS and A level mathematics practice in England (the GCE subject content).