Implicit Differentiation

    In this topic, the procedure of implicit differentiation is outlined and many examples are given. Proofs of the derivative formulas for the inverse trigonometric functions are provided and several examples of using them are given. Also detailed is the logarithmic differentiation procedure which simplifies the process of taking derivatives of equations involving products and quotients.  

    Finding the slope of a tangent line is a local process; for example, a circle  locally around a point, can have a tangent line even though it is not a function. In fact, every circle has a tangent line at every point. The following process allows us to find derivatives of more general curves (not just functions); and in particular for an implicitly defined function. Notice that the process relies heavily on the chain rule.

Proposition (Implicit Differentiation) Suppose that is a given equation involving both and ; and that exists at Then can be found using the following procedure (called implicit differentiation):

    (i) Using the chain rule where appropriate, differentiate both sides of the equation with respect to
    
    (ii) If possible, solve the differentiated equation algebraically for and evaluate at
    

Example (Implicit Differentiation) Use implicit differentiation.to find given

    Solution. We will use implicit differentiation, and in doing so we use the chain rule on the right hand and the product rule together with the chain rule on the left hand side of the equation:













Example (Implicit Differentiation) Use implicit differentiation.to find the tangent to the folium of Descartes at

    Solution. Using implicit differentiation we have,





So the tangent line at is



which simplifies to The graph of the folium and the tangent line are shown:
    



Example (Implicit Differentiation) Use implicit differentiation.to find the tangent to the lemniscate of Bernoulli at

    Solution. Using implicit differentiation we have,





So the tangent line at is



which simplies to The graph of the folium and the tangent line are shown:




Example (Implicit Differentiation) Use implicit differentiation.to find all points on the lemniscate of Bernoulli where the tangent line is horizontal.

    Solution. Using implicit differentiation we have,





we need to find all where Clearly, the point is ruled out and so ; that is Using with the original we see also. Therefore, and so and


    
Example (Implicit Differentiation) Use implicit differentiation.to find two points on the curve whose equation is where the tangent line is vertical.

    Solution. Using implicit differentiation we determine,



and so,



Since we want vertical tangent lines we need that is, and with the original equation this means;



which is solved as So the points where the tangent line is vertical are and

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