## Calculus Tutorial

*Basic derivatives (The power rule)*

#### Intro

Differentiating can be a pretty daunting task. Although there are many different rules to follow for more advanced problems, the power rule is the basic rule of derivatives.

The derivative of a function provides the equation of the slope of the original function. Each point on the derivative graph gives the instantaneous rate of change of the respective point on the original graph.

#### Sample Problem

Derive the following equations:

a) f(x) = x^3

b) g(x) = 2x^7

c) h(x) = x^-1

d) l(x) = 9x

e) k(x) = 9x^2 + 4x

#### Solution

To begin, there is an equation that you can always resort back to when dealing with the power rule.

**The power rule**

*f (prime) (x) is usually written as “f followed by a superscript roman numeral 1 then (x)”
*

f(x) = x^n, n belonging to all real numbers

f (prime) (x) = nx^(n-1)

*Things to note: When deriving x, x will become 1, rather than x^0*

a)

f(x) = x^3

f (prime) (x) = (3)n^(3-1)

= 3n^2

b)

g(x) = 2x^7

g (prime) (x) = (7)2x^(7-1)

= 14x^6

c)

h(x) = x^-1

h (prime) (x) = (-1)x^(-1-1)

= -x^-2

d)

l(x) = 9x

l (prime) (x) = (9) (1)

= 9

e)

k(x) = 9x^2 + 4x

k (prime) (x) = (2)9x^(2-1) + 4(1)

= 18x + 4

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