A tangent or normal to a curve is a line, taking the formwhereis the gradient and is the intercept. Given a functionwe can find the gradient atby finding the gradient functionand substituting the valueinto this expression. Sometimes however we don’t havesois not given explicitly as a function ofIn these cases typically we have to differentiate implicitly and findas a function of bothandand then substitute a pointinto the expression forto find the gradient at that point. Finally substitute into the equationto find the equation of the line.

Example: Find the equation of the tangent to the curveat the point

We differentiate implicitly to getThe gradient at the point is

Example: Find the equation of the tangent to the curveat the point

We differentiate implicitly to getWe have to make the subject of this equation.

The gradient at the point is

Example: Find the equation of the normal to the curveat the point

We differentiate implicitly to getWe have to make the subject of this equation.

The gradient at the point is