Derivative of Inverse

    First we discuss implicit differentiation and then we illustrate the derivative of an inverse function theorem. We finish by giving several examples of logarithmic differentiation.  

(1) 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
    

(2) 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:













(3) 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:
    



(4) 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 simplifies to The graph of the folium and the tangent line are shown:




(5) 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


    
(6) 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

(7) Proposition (Derivative of Inverse) If has an interval as domain and exists and is never zero on then is differentiable at every point in its domain. The value of at a point in the domain of is the reciprocal of the value of at given by

(8) Example (Derivative of Inverse) Let Sketch the graph of and state whether or not  the graph of passes the horizontal line test. If so, find a rule for and then use it to find . Verify the formula

(9) Example (Derivative of Inverse) Let Without finding a rule for determine

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

    (i) Apply a logarithmic function with the appropriate base to both sides.
    
    (ii) Use properties of logarithms to simplify.

    (iii) Differentiate both sides of the equation with respect to
    
    (iv) If possible, solve the differentiated equation algebraically for
    

(11) Example (Logarithmic Differentiation) Use logarithmic differentiation to find given

    Solution. Using the natural logarithmic function, we have



and applying implicit differentiation we have,





as desired.

(12) Example (Logarithmic Differentiation) Use logarithmic differentiation to find given

    Solution. Using the natural logarithmic function, we have



and applying implicit differentiation we have,





as desired.

• print this page
• pages linking here
• related pages

Cite this as:
Derivative Of Inverse
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/derivative-of-inverse.html