MATH 111 Lecture Notes - Lecture 12: Implicit Function, Chain Rule
9 Jun 2015
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1476- lecture 12 - implicit differentiation and derivatives of logarithmic functions. As approaches, sin ~ . (d/dx) [ f ( g(x) )] = f ( g(x) ) g"(x) dx. Keep deriving the inner functions until you arrive at dx. Some equations cannot be easily differentiated with respect to x. Instead of differentiating like normal, whenever you differentiate y, multiply it by dy (instead of, for example: f(x)= x2; f (x)= 2x * dx ) (d/dx) (x3 + y3 ) = (d/dx) (6xy) (d/dx) (2x2dx + 2y2 dy) = (d/dx) (y*6 dx + 6x*dy/dx) 2x2 + 2y2dy/dx = 6y + 6x*dy/dx. 2y2dy/dx - 6x*dy/dx = 6y - 2x2. Dy/dx (2y2 - 6x) = 6y - 2x2. Dy/dx= (6y - 2x2) / (2y2 - 6x) Dy/dx = (3y-x2) / (y2 - 3x) As you get more comfortable with implicit differentiation, you can combine several steps into one: Differentiate both sides of the equation with regards to x.
