Rules for Derivatives
Derivatives (Dx): β’ In this tutorial we will use Dx for the derivative ( ). Dx indicates that we are taking the derivative with respect to x. ( ) is another symbolπ π for representing a derivative. π π π π β’ The derivative represents the slopeβ² of the function at some x, and slope represents a rate of change at that point. ππ π₯π₯ β’ The derivative (Dx) of a constant (c) is zero. . Ie: y = 3 since y is the same for any x, the slope is zero (horizontal line)
Power Rule: The fundamental tool for finding the Dx of f(x)
β’ multiply the exponent times the coefficient of x and then reduce the exponent by 1
2 2 Ex: [ ] ( ) = 3 = 3 1 2 2 [ ] ( ) = [ ππ + ] β² ( ) = 3 = 3 π¦π¦π¦π¦π¦π¦π¦π¦π¦π¦π¦π¦ π«π«π«π« ππ β ππ π₯π₯ π₯π₯ ππππ π₯π₯ ππ β² ππβ ππ β² π·π·π·π· π₯π₯ οΏ½β―β―β―οΏ½ ππ π₯π₯ πππ₯π₯ * [dx representsπ«π« theπ«π« ππderivativeππ β ofππ whatπ₯π₯ is insideπ₯π₯ ππππ (x), whichπ₯π₯ is usually 1 for simple functions, the dx must always be considered and is always there, even if it is only 1]
Sum Rule: The Dx of a sum is equal to the sum of the Dxβs
[ ( ) + ( )] ( ) + ( ) β² β² Ex: [3 2 + 2 + 3] (3 2) + (2 ) + (3π·π·)π·π·=ππ6π₯π₯ + 2ππ+π₯π₯0 βΉ ππ π₯π₯ ππ π₯π₯ β² β² β² Constant Coefficientπ·π·π·π· π₯π₯ Rule:π₯π₯ Theβ Dxππ ofπ₯π₯ a variableππ π₯π₯ with πΉπΉa constantπ₯π₯ coefficient is equal to the constant times the Dx. The constant can be initially removed from the derivation.
[3 2] = 3( [ 2])
Ex: [ln(4) 2] = ln(4) [ 2] = ln(4) 2 = 2 ln(4) =π·π·lnπ·π·(4π₯π₯)2 = ln(π·π·16π·π· )π₯π₯
Chain Rule:π·π·π·π· There isπ₯π₯ nothing newπ·π·π·π· hereπ₯π₯ other thanβ theπ₯π₯ dx is now somethingπ₯π₯ otherπ₯π₯ than 1. π₯π₯ The dx represents the Dx of the inside function g(x). It is called a chain rule because you have to consider the dx as not being 1 and take the Dx of the inside also.
[ ( ) ( ) = ( ) ( ) β² β² β² Ex: Dx (sin(3x)) = cos(3x) dx* = 3 cos(3x) π·π· π·π· * ππ[dxοΏ½ππ isπ₯π₯ gβ(οΏ½ 3xβΉ) ππ= οΏ½3ππ] π₯π₯ οΏ½ππππ ππ οΏ½ππ π₯π₯ οΏ½ππ π₯π₯
* [the dx here is gβ(x)]
Ex: Dx [(3x2+2)2] = 2(3x2+2) dx* = 2( 3x2+2 ) (6x) = (6x2 + 4)(6x) = 36x3 + 24x
*[dx is Dx (3x2 + 2) = 6x] notice we used the Power Rule along with the Chain Rule
James S__ Jun 2010 r6 Rules for Derivatives
U-sub: This is when you let some letter equal the whole inside quantity. It can be very useful in a Chain Rule situation. Ex: Dx [(sin(x))3] βΊIf we let U = sin(x) then du = cos(x) Now we have: Dx [U3] = 3U2du 3[sin(x)]2[cos(x)] βΊsubstituteβΉ back in for U and du
ProductβΉ Rule: The Dx of a product is equal to the sum of the products Dx of each factor times the other factor. [ ( ) ( )] [ ( ) ( ) + ( ) ( ) ]
β² β² π·π·π·π· ππ π₯π₯ β’ ππ π₯π₯ βΉ ππ π₯π₯ ππππ β’ ππ π₯π₯ ππ π₯π₯ β’ ππ π₯π₯ ππππ Ex: 3 2 = 6 ( ) + (3 2) π₯π₯ π₯π₯ π₯π₯ Quotient Ruleπ₯π₯: ππDx (numerator)π₯π₯ ππ timesπ₯π₯ ππ the denominator minus Dx (denominator) times the numerator, divided by the denominator squared. This is a variation of the Product Rule.
( ) ( ) ( ) ( ) ( )
( ) β² [ ( )]2 β² ππ π₯π₯ ππ π₯π₯ ππ π₯π₯ β ππ π₯π₯ ππ π₯π₯ π·π·π·π· οΏ½ οΏ½ βΉ ππ π₯π₯ ππ π₯π₯ sin ( ) cos ( )(3 ) sin (3 )(3) 3 ( ) 3sin (3 ) ( ) sin (3 ) = = = Ex: 3 (3 )2 9 2 3 2 π₯π₯ π₯π₯ π₯π₯ β π₯π₯ π₯π₯π₯π₯π₯π₯π₯π₯ π₯π₯ β π₯π₯ π₯π₯π₯π₯π₯π₯π₯π₯ π₯π₯ β π₯π₯
π·π·π·π· οΏ½ π₯π₯ οΏ½ π₯π₯ π₯π₯ π₯π₯ Special Rules: 1 1 1 β’ [ln( )] = = [ ( ) ] = ( ) ( ) π·π·π·π· ππππππππ π₯π₯ ππ ππππ π·π·π·π· π₯π₯ π₯π₯ ππππ ππ ππππ 1 π₯π₯ cos ( ) [ln(sin( ))] = cos( ) = = cot( ) π₯π₯ππππ ππ Ex: sin ( ) ( ) sin ( ) π₯π₯
π·π·π·π· π₯π₯ π₯π₯1 ππππ ππ π₯π₯ 3 π₯π₯ π₯π₯ [log(3 2)] = (6 ) = Ex: (3 2)( 10) ( )( 10)
π·π·π·π· π₯π₯ π₯π₯ ππππ π₯π₯ π₯π₯ ππππ β’ [ ] = ln( ) [ ] = (ln( )) π₯π₯ π₯π₯ π₯π₯ π₯π₯ Ex:π·π·π·π· Dxππ [3e4x]ππ = 3ππ[ππ(e4x )[ln(e)]ππ (4)] = 12(e4x) π·π·π·π· ππ ππ ππ ππππ
2 2 Ex: [13 +5 ] = (13 +5 )[ln(13)](2 + 5) π₯π₯ π₯π₯ π₯π₯ π₯π₯ π·π·π·π· π₯π₯
James S__ Jun 2010 r6