Rules for Derivatives
James S__ Jun 2010 r6
Derivatives (Dx):
• In this tutorial we will use Dx for the derivative (𝒅𝒅
𝒅𝒅𝒅𝒅). Dx indicates that we are taking the
derivative with respect to x. 𝑓𝑓′(𝑥𝑥) is another symbol for representing a derivative.
• The derivative represents the slope of the function at some x, and slope represents a rate of
change at that point.
• 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)
• multiply the exponent times the coefficient of x and then reduce the exponent by 1
Ex: 𝑫𝑫𝒅𝒅 [𝒅𝒅𝟑𝟑]→ 𝑓𝑓′(𝑥𝑥)= 3𝑥𝑥2 𝑑𝑑𝑥𝑥 = 3𝑥𝑥2
𝑫𝑫𝒅𝒅 [𝒅𝒅𝟑𝟑+𝟓𝟓]→ 𝑓𝑓′(𝑥𝑥)= 3𝑥𝑥2 𝑑𝑑𝑥𝑥 = 3𝑥𝑥2
* [dx represents the derivative of what is inside (x), which 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 Dx’s
Ex: 𝐷𝐷𝑥𝑥 [3𝑥𝑥2+ 2𝑥𝑥+ 3]→ 𝑓𝑓′(3𝑥𝑥2)+𝑓𝑓′(2𝑥𝑥)+ 𝐹𝐹′(3)= 6𝑥𝑥+ 2 + 0
Constant Coefficient Rule: The Dx of a variable with a constant coefficient is equal to the
constant times the Dx. The constant can be initially removed from the derivation.
Ex: 𝐷𝐷𝑥𝑥[ln(4)𝑥𝑥2]=ln(4)𝐷𝐷𝑥𝑥[𝑥𝑥2]=ln(4)∗2𝑥𝑥= 2 ln(4)𝑥𝑥=ln(4)2𝑥𝑥=ln(16)𝑥𝑥
Chain Rule: There is nothing new here other than the dx is now something other than 1. 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.
Ex: Dx (sin(3x)) = cos(3x) dx* = 3 cos(3x) * [dx is g’(3x) = 3]
* [the dx here is g’(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
𝐷𝐷𝑥𝑥 [𝑥𝑥𝑛𝑛]𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑑𝑑𝑦𝑦
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𝑓𝑓′(𝑥𝑥)= 𝑛𝑛𝑥𝑥𝑛𝑛−1
𝐷𝐷𝑥𝑥[𝑓𝑓(𝑥𝑥)+ 𝑔𝑔(𝑥𝑥)] ⟹ 𝑓𝑓′(𝑥𝑥)+ 𝑔𝑔′(𝑥𝑥)
𝐷𝐷𝑥𝑥[3𝑥𝑥2]= 3(𝐷𝐷𝑥𝑥[𝑥𝑥2])
𝐷𝐷𝑥𝑥 [𝑓𝑓�𝑔𝑔(𝑥𝑥)�⟹𝑓𝑓′�𝑔𝑔(𝑥𝑥)�𝑑𝑑𝑥𝑥 =𝑓𝑓′�𝑔𝑔(𝑥𝑥)�𝑔𝑔′(𝑥𝑥)
