Differentiation of Implicit Functions

Brian Knight and Roger Adams
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Brian Knight: Goldsmiths’ College
Roger Adams: Thames Polytechnic

Chapter 6 in Calculus I, 1975, pp 44-49 from Springer

Abstract: Abstract If we are not given y explicitly as an expression in x alone, but are given instead an equation in x and y, such as: (6.1) x y + sin y = 0 $$ xy + \sin y = 0 $$ then y is said to be defined implicitly by the equation. In this case, the easiest way to find dy/dx is to differentiate the whole equation through term by term. Hence, differentiating xy with respect to x, we get by the product rule: d ( x y ) d x = 1 ⋅ y + x ⋅ d y d x $$ \frac{{d\left( {xy} \right)}} {{dx}} = 1 \cdot y + x \cdot \frac{{dy}} {{dx}} $$ and using the function of a function rule for sin y: d ( sin y ) d x + d ( sin y ) d y ⋅ d y d x = cos y d y d x $$\frac{{d\left( {\sin y} \right)}} {{dx}} + \frac{{d\left( {\sin y} \right)}} {{dy}} \cdot \frac{{dy}} {{dx}} = \cos y\frac{{dy}} {{dx}}$$

Keywords: Inverse Function; Product Rule; Sine Function; Class Discussion; Function Rule (search for similar items in EconPapers)
Date: 1975
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DOI: 10.1007/978-1-4615-6594-9_6

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