In the previous topic we saw that the gradient of a curve is given by finding the gradient of the tangent to the curve at that point.

By differentiating and finding the derived or gradient function we obtain an equation which will give the gradient at any point on the curve.

### Finding the gradient at a point on a curve

To find the gradient at a particular point on the curve of the function f(x):

Step 1Differentiate the function to find f '(x)Step 2Substitute the x value of the point into f '(x)

Find the gradient of the function f(x) = x Differentiate:
Gradient at x = -1 is
The diagram shows this answer. |

Further example

### Finding the Equation of the Tangent to a Curve

Once the **gradient** of a tangent to a curve has been found, the formula** **

**y − y _{1} = m(x − x_{1}) **can be used to find the

**equation**of the tangent.

**Example**

Find the equation of the tangent to the curve f(x) = x^{2} + 5x at the point (3, 24)

Differentiating f '(x) = 2x + 5

Substitute x = 3

f '(3) = 2 x 3 + 5 = 11

Using y − y

_{1}= m(x − x_{1})Equation is y − 24 = 11(x − 3)

Re-arranging y − 24 = 11x − 33

y = 11x − 9

**The equation of the tangent is y = 11x − 9**