Gradient of a Curve at a Point

Worked example

1. 1

   $y=x^2,x=1\Rightarrow (1,1)$ — The curve and the point. Put x = 1 into the curve to get y = 1, and mark this point on the graph.

2. 2

   $\text{tangent through }(1,1)$ — Lay a ruler so it just touches the curve at the point, then draw the tangent long, with the curve not crossing it.

3. 3

   $(0,-1)\text{and}(2,3)$ — Read two points on the tangent, far apart. Far apart points give a more accurate gradient.

4. 4

   $\text{up}=3-(-1)=4$ — Up is the difference in y. The brackets round −1 stop a sign slip, because taking away a negative adds.

5. 5

   $\text{across}=2-0=2$ — Across is the difference in x, taken in the same order as the y values.

6. 6

   $\frac{4}{2}=2$ — Gradient is up ÷ across. The line goes up to the right, so the gradient is positive.

Worked example

1. 1

   $y=x^2\Rightarrow \frac{dy}{dx}=2x$ — Differentiate: the power 2 comes down in front and the power drops by 1.

2. 2

   $x=1:2(1)=2$ — Put x = 1 into dy/dx. This is the same 2 the tangent gave, but exact.

Worked example

1. 1

   $y=x^3-5x^2+4,x=-1$ — The curve and the x value. The x value is negative.

2. 2

   $x^3\to 3x^2$ — The power 3 comes down, and the power drops from 3 to 2.

3. 3

   $-5x^2\to -10x$ — 2 × (−5) = −10, and the power drops from 2 to 1.

4. 4

   $4\to 0$ — A number on its own is flat, so it has no slope and gives 0.

5. 5

   $\frac{dy}{dx}=3x^2-10x$ — Put the three results together.

6. 6

   $3(-1{)}^2-10(-1)$ — Put −1 in brackets wherever x was.

7. 7

   $3(1)-10(-1)$ — (−1)² = 1, because a negative times a negative is positive.

8. 8

   $3+10=13$ — −10 × (−1) = +10. The gradient at x = −1 is 13.

Worked example

1. 1

   $y=2x^3-3x^2-12x+4$ — The curve.

2. 2

   $\frac{dy}{dx}=6x^2-6x-12$ — Differentiate each term: 2x³ → 6x², −3x² → −6x, −12x → −12, and 4 → 0.

3. 3

   $6x^2-6x-12<0$ — A negative gradient means dy/dx is below 0.

4. 4

   $x^2-x-2<0$ — Divide every term by 6. It is positive, so the < stays the same.

5. 5

   $(x-2)(x+1)=0$ — Factorise. You need two numbers that multiply to −2 and add to −1: −2 and +1.

6. 6

   $x=2\text{or}x=-1$ — Each bracket equal to 0 gives a root. These are the places where the gradient is exactly 0.

7. 7

   $\text{U-shape, roots at }-1\text{ and }2$ — The x² term is positive, so sketch a U. It is below the axis between its two roots.

8. 8

   $-1<x<2$ — Below the axis means < 0. Check: at x = 0 the gradient is −12, which is negative.