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Linearity

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Miggy Ceballo
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42 views3 pages

Linearity

Uploaded by

Miggy Ceballo
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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Linearity

Product rule

Reciprocal rule

Quotient rule

Chain rule

where (f g)(x) is defined as f(g(x))

For higher derivatives the chain rule is given by Faà di Bruno's formula (below is the combinatoric form):

Derivative of inverse function

for any differentiable function f of a real argument and with real values, when the indicated compositions and
inverses exist.

Generalized power rule

Derivative of implicit function


If implicit function y(x) is defined as F(x,y(x)) = 0
then

Derivative of parametrically defined function


If a function y(x) defined parametrically
then

Derivative of complex function.


For complex function z(t) = F(u(t),v(t))
the full derivative is

If z(x) = F(x,y(x))
Then full derivative is

Derivatives of exponential and logarithmic functions

note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.

the equation above is also true for all c but yields a complex number.

The derivative of the natural logarithm with a generalised functional argument f(x) is

By applying the change-of-base rule, the derivative for other bases is


Derivatives of trigonometric functions

Derivatives of hyperbolic functions

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