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Recently Added Math Formulas
·
Integrals of Trigonometric Functions
·
Integrals of Hyperbolic Functions
·
Integrals of Exponential and Logarithmic Functions
·
Integrals of Simple Functions
·
Integral (Indefinite)
Additional Formulas
·
Derivatives Basic
·
Differentiation Rules
·
Derivatives Functions
·
Derivatives of Simple Functions
·
Derivatives of Exponential and Logarithmic Functions
·
Derivatives of Hyperbolic Functions
·
Derivatives of Trigonometric Functions
·
Integral (Definite)
·
Integral (Indefinite)
·
Integrals of Simple Functions
Current Location
>
Math Formulas
>
Calculus
> Differentiation Rules
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. x
^{2}
+ 3xy + y
^{2}
= 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
x
^{3}
3y
^{2}
=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{a
^{x}
} = a
^{x}
lna
Recall that
a
^{x}
a
^{x}
= a
^{x+x}
= a
^{2x}
and
a
^{x}
b
^{x}
= (ab)
^{x }
Recall that
lnA – lnB = ln(A/B)
Example 4:
Differentiate
Factor 2
x
and (x
^{2}
– 1)
^{2}
from the numerator
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