Difference between revisions of "Differentiation Rules"
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| − | Differentiation | + | '''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== | ==Basic Rules== | ||
'''Derivative of a Constant:''' | '''Derivative of a Constant:''' | ||
| − | If <math>y=c</math> | + | If <math>y(x)=c</math> is a constant function then <math>\frac{dy}{dx} = 0</math>. |
'''Sum Rule:''' | '''Sum Rule:''' | ||
| − | 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:''' | '''Product Rule:''' | ||
| − | If <math>y(x) = u(x) | + | 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:''' | '''Quotient Rule:''' | ||
| − | If <math>y( | + | 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:''' | '''Chain Rule:''' | ||
| − | If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv} | + | If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>. |
| − | ''' | + | '''Power Rule:''' |
| − | If <math>y(x) = u(x^n | + | If <math>y(x) = (u(x))^n</math> then <math>\frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx}</math>. For [[integer]] <math>n</math> this is just a consequence of the product and quotient rules and [[induction]], but it can also be proven for all [[real number]]s <math>n</math>, e.g. by using the [[Extended Binomial Theorem]]. |
==Derivatives of Trig Functions== | ==Derivatives of Trig Functions== | ||
Revision as of 08:43, 17 August 2009
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.
