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A comprehensive tutorial on the rules for derivatives, including the power rule, sum rule, constant coefficient rule, chain rule, u-sub, product rule, quotient rule, and special rules. It covers various examples and explanations to help understand the concepts.
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James S__ Jun 2010 r
𝒅𝒅
𝒅𝒅𝒅𝒅
). Dx indicates that we are taking the
derivative with respect to x. 𝑓𝑓
′
(𝑥𝑥) is another symbol for representing a derivative.
change at that point.
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)
Ex: 𝑫𝑫𝒅𝒅 [𝒅𝒅
𝟑𝟑
′
2
2
𝟑𝟑
′
2
2
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
2
′
2
′
′
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) 𝑥𝑥
2
] = ln(4) 𝐷𝐷𝑥𝑥
2
] = ln(4) ∗ 2 𝑥𝑥 = 2 ln(4) 𝑥𝑥 = ln(4)
2
𝑥𝑥 = 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]
Ex: Dx [(3x
2
2
] = 2(3x
2
+2) dx* = 2( 3x
2
+2 ) (6x) = (6x
2
3
*[dx is Dx (3x
2
𝑛𝑛
𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑑𝑑𝑦𝑦
′
𝑛𝑛 − 1
′
′
2
2
′
′
′
James S__ Jun 2010 r
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
Now we have: Dx [U
] ►If we let U = sin(x) ⟹ then du = cos(x)
3
2
⟹ 3[sin(x)]
du
2
[cos(x)] ►substitute back in for U and du
Product Rule: The Dx of a product is equal to the sum of the products Dx of each factor
times the other factor.
Ex: 3 𝑥𝑥
2
𝑥𝑥
𝑥𝑥
2
𝑥𝑥
Quotient Rule : Dx (numerator) times the denominator minus Dx (denominator) times the
numerator, divided by the denominator squared. This is a variation of the Product Rule.
Ex: 𝐷𝐷𝑥𝑥 �
sin (𝑥𝑥)
3 𝑥𝑥
cos (𝑥𝑥
)( 𝑥𝑥
) −sin (3𝑥𝑥
)(3)
(3𝑥𝑥)
2
3 𝑥𝑥𝑥𝑥𝑥𝑥𝑦𝑦 (𝑥𝑥)−3sin (3𝑥𝑥)
9 𝑥𝑥
2
𝑥𝑥𝑥𝑥𝑥𝑥𝑦𝑦 (𝑥𝑥)−sin (3𝑥𝑥)
3 𝑥𝑥
2
Special Rules:
1
𝑥𝑥 𝑦𝑦𝑛𝑛 (𝑦𝑦)
1
𝑥𝑥
Ex: 𝐷𝐷𝑥𝑥
[ln(sin( 𝑥𝑥
1
sin (𝑥𝑥)𝑦𝑦𝑛𝑛 (𝑦𝑦)
cos(𝑥𝑥
cos (𝑥𝑥)
sin (𝑥𝑥)
= cot(𝑥𝑥)
Ex: 𝐷𝐷𝑥𝑥 [log(3𝑥𝑥
2
1
(3𝑥𝑥
2
)(𝑦𝑦𝑛𝑛 10)
3
(𝑥𝑥)(𝑦𝑦𝑛𝑛 10)
𝑥𝑥
𝑥𝑥
Ex: Dx [3e
4x
] = 3[(e
4x
)ln(e)] = 12(e
4x
Ex: 𝐷𝐷𝑥𝑥 [
𝑥𝑥
2
+5𝑥𝑥
𝑥𝑥
2
+5𝑥𝑥
)[ln(13)](2𝑥𝑥 + 5)
′
′
′
′
2
𝑏𝑏
𝑏𝑏
𝑥𝑥
𝑥𝑥