This chapter explores implicit differentiation. The chapter covers differentiating a function defined implicitly; Finding equations of tangents and normals to curves defined implicitly. Before attempting this chapter you must have prior knowledge of basic differentiation, and tangent and normals. The functions we’ve explored so far are in the form of; Functions can also be expressed implicitly. These are functions which cannot be rearranged into form; An example is that of a circle shown below. …this can be rearranged into the form y=f(x) but it would look too complicated. Let’s try to do that; It is better if we leave it as; Below are other examples; You may have noticed that the functions are a mixture of x’s and y’s.
Differentiating implicit functionsBelow 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 y2;
Example…we first use the product rule; …we then use the chain rule to differentiate y3 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 tangentWe 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 questionThe 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?