Difference between revisions of "Differentiation Rules"
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If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>. | If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>. | ||
| − | + | [[Product Rule]]: | |
If <math>y(x) = u(x) \cdot v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>. | If <math>y(x) = u(x) \cdot v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>. | ||
| − | + | [[Quotient Rule]]: | |
If <math>y(x) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>. | If <math>y(x) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>. | ||
| − | + | [[Chain Rule]]: | |
If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>. | If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>. | ||
Revision as of 23:53, 25 March 2018
Differentiation rules are rules (actually, theorems) used to compute the derivative of a function in calculus. In what follows, all functions are assumed to be differentiable.
Basic Rules
Derivative of a Constant:
If 
is a constant function then 
.
Sum Rule:
If 
then 
.
Product Rule:
If 
then 
.
Quotient Rule:
If 
then 
.
Chain Rule:
If 
then 
.
Power Rule:
If 
then 
. For integer 
this is just a consequence of the product and quotient rules and induction, but it can also be proven for all real numbers , e.g. by using the extended Binomial Theorem.
Derivatives of Trig Functions
Derivative of Sine
If 
, then 
.
Derivative of Cosine
If 
, then 
.
Derivative of Tangent
If 
, then 
. Note that this follows from the Quotient Rule.
