## Differentiating implicit functions

Below is a function that has been defined implicitly; Suppose we wanted to find dy/dx. Notice this is different from the usual y=f(x). It simply means we find the differential of the function as in; ...for the function in question; ...it means we differentiate each term one at a time so we have; ...we know that; Now we have to find the differential of xy; ...we’re trying to find; ...we can use the product rule so we get; ...so now we have; ...we can rearrange it to make dy/dx the subject; Above we have managed to find dy/dx Below are more examples;### Example

...we have; ...we can use the chain rule to differentiate y^{2};

### Example

...we first use the product rule; ...we then use the chain rule to differentiate y^{3}according to y;

### Example

...we use the product rule first; we use the chain rule for; ...thus we get;### Example

...we have to use the chain rule on sin y;### Example

[ ...we’re trying to find; ...we differentiate each part separately; We do each part using the chain rule;## Finding the tangent

We can also find tangents and normals to curves defined implicitly, for example; Finding the equation of the tangent to the curve at the point (1, 4) To find the gradient we differentiate, that is; ...we get; [] ...we substitute in the point (1, 4) to get; Now we can use; ...we substitute in to get; The answer for the equation of the tangent is;## Exam question

The equation of a circle is given as;- What is the centre and radius of the circle?
- Find the coordinates where the circle crosses the line x=4
- Find the equations of the normals to the circle at these points
- Where do the normals interest?

## Differentiating implicit functions

Below is a function that has been defined implicitly; Suppose we wanted to find dy/dx. Notice this is different from the usual y=f(x). It simply means we find the differential of the function as in; ...for the function in question; ...it means we differentiate each term one at a time so we have; ...we know that; Now we have to find the differential of xy; ...we’re trying to find; ...we can use the product rule so we get; ...so now we have; ...we can rearrange it to make dy/dx the subject; Above we have managed to find dy/dx Below are more examples;### Example

...we have; ...we can use the chain rule to differentiate y^{2};

### Example

...we first use the product rule; ...we then use the chain rule to differentiate y^{3}according to y;

### Example

...we use the product rule first; we use the chain rule for; ...thus we get;### Example

...we have to use the chain rule on sin y;### Example

[ ...we’re trying to find; ...we differentiate each part separately; We do each part using the chain rule;## Finding the tangent

We can also find tangents and normals to curves defined implicitly, for example; Finding the equation of the tangent to the curve at the point (1, 4) To find the gradient we differentiate, that is; ...we get; [] ...we substitute in the point (1, 4) to get; Now we can use; ...we substitute in to get; The answer for the equation of the tangent is;## Exam question

The equation of a circle is given as;- What is the centre and radius of the circle?
- Find the coordinates where the circle crosses the line x=4
- Find the equations of the normals to the circle at these points
- Where do the normals interest?