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Differentiation Rules

Chain Rule


Inverse Function


Linearity
, where c is a constant



Product Rule


Quotient Rule


Reciprocal Rule



Differentiation of Implicit Functions
If an equation is expressed as y = f(x), then y is said to be the explicit function of x. However if y is connected with x by an expression f (x, y) = 0, then y is said to an implicit function of x. For e.g. x2 + 3xy + y2 = 0.

For differentiating the implicit function we differentiate each term with respect to x keeping in mind that if there is a term contains powers of y, we differentiate first with respect to y and then multiply it by
dy/dx to get its differentiation with respect to x.

Example 1:




Example 2: Calculate dy/dx for x3-3y2=21
Differentiating both side of equation with respect to x, and using The Sum Rule:



Now using The Chain Rule which leads to:



Example 3: Differentiate


Recall that D{ax} = axlna



Recall that axax = ax+x = a2x and axbx = (ab)x



Recall that lnA – lnB = ln(A/B)

 



Example 4: Differentiate



Factor 2x and (x2 – 1)2 from the numerator




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