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Calculus

Basic Differentiation Rules

CONSTANT FUNCTION RULE 

A constant function is of the form \( f(x) = b \) where is some real number. The derivative of ALL constant functions is always 0. 

Consider \( f(x) = 5 \). We will have that \( f'(x) = 0 \). 

Why is this so? 

Suppose you are driving a car from point A to point B, and the whole time you are driving 40km/h. Your speed does not change, and so the car is going at a constant pace. If you were to calculate the rate of change of your speed, you would get zero - because there IS no change in your speed! 

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POWER RULE 

The power rule applies to functions that are of the form \( f(x) = x^n \). We have that the derivative of these types of functions will always be of the form \(f'(x) = nx^{n-1} \).

Consider \( f(x) = x^5 \). 

                                                                                   \( \Longrightarrow f'(x) = 5x^{5-1} \)

\[ \Longrightarrow f'(x) = 5x^4 \]

 

CONSTANT MULTIPLE RULE 

The constant multiple rule is a variation of the power rule, and applies to functions of the form \( f(x) = cx^n \), where \( c \) is some real number. We have that the derivative of these types of functions is of the form \( f'(x) = (c)(n)x^{n-1} \). 

Consider \( f(x) = 4x^6 \).

                                                                                  \( \Longrightarrow f'(x) = (4)(6)x^5  \)

\[ \Longrightarrow f'(x) = 24x^5 \]

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SUM RULE

The sum rule applies to functions with more than one term. To determine the derivative of these functions, we take the derivatives of each of the individual terms. If we have that \( f(x) = g(x) + h(x) \), where \( g(x), h(x) \) are functions, then \( f'(x) = g'(x) + h'(x) \). 

Consider \( f(x) = 2x^3 + x^4 \).

                                                                              \( \Longrightarrow f'(x) = (2)(3)x^2 + 4x^3 \) 

\[ \Longrightarrow f'(x) = 6x^2 + 4x^3 \]

 

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