Finding a Tangent
Using the derivative to find a tangent
- At any point on a curve, the tangent is the line that touches the point and has the same gradient as the curve at that point
- When given a curve, you can find the equation of the tangent to the curve at the point by:
- Finding the derivative (gradient) of the curve at point
- This is also the gradient of the tangent line
- You can find this by differentiating the equation of the curve, and substituting in
- Substituting the value of the gradient into the equation of the tangent, in the form
- To find the full equation of the tangent, substitute in the point as and and solve to find
- You could alternatively use the form for the equation of a line where is the point and is still the gradient
- Finding the derivative (gradient) of the curve at point
- Sometimes, you may not be told the full coordinate; just the -value
- In this case, substitute the -value into the equation of the curve (not the derivative) to find the full coordinate, and then follow the method above
Exam Tip
- A good sketch of the curve and the tangent at a point can help you spot if the tangent will have a positive or negative gradient; helping you to check your answer
Worked example
Work out the equation of the tangent to the curve at the point where .
Write your answer in the form .
Find the derivative of the curve.
To find the gradient of the curve at the point where substitute into the derivative of the curve.
This is the same as the gradient of the tangent to the curve at the point where , so the equation of the line is in the form
To find the value of we will need to know the full coordinate at the point where We can find this by substituting into the equation for the curve (be careful to substitute it into the original equation and not your differentiated version).
Simplify to find the value of and hence, the full coordinate.
Substitute and into the equation of the line.
Solve this equation to find
Write out the equation of the tangent to the curve in the form given in the question.