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# Basic Rules of Differentiation Study Help

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Updated on Oct 1, 2011

## Basic Rules of Differentiation

Using the limit definition to find derivatives can be very tedious. Luckily, there are many shortcuts available. For example, if function f is a constant, like f (x) = 5 or f (x) = 18, then f' (x) = 0. This can be proven for all constants c at the same time in the following manner.

If:

f (x) = c

then:

All of the general rules in this chapter can be proven in such a manner, using the limit definition of the derivative, though we shall not bother to do so. The first rule is the Constant Rule, which says that if f (x) = c where c is a constant, then f' (x) = 0.

Before we go any further, a word needs to be said about notation. The concept of the derivative was discovered by both Isaac Newton and Gottfried Leibniz. Newton would put a dot over an object to represent its derivative, much like the way f' (x) represents the derivative of f (x). Leibniz would write the derivative of y (where x is the variable) as . Newton's notation is certainly more convenient, but Leibniz's enables us to represent "take the derivative of something" as (something). Thus, if y = f (x) , then = (f (x)) = f' (x).Using Leibniz's notation, the Constant Rule where c is a constant is (c) = 0.

We will use both forms of the Constant Rule, depending on the situation. The next rule is the Power Rule, which is stated: (xn) = n · xn – 1. This rule says "multiply the exponent in front and then subtract one from it."

### Constant Rule

If f (x) = c where c is a constant, then f'(x) = 0.

And, using Leibnez's notation, if c is a constant, then (c) = 0.

### Power Rule

#### Example 1

Differentiate f(x) = x2.

#### Solution 1

f' (x) = 2x2–1 = 2x1 = 2x

#### Example 2

Differentiate y = x8.

= 8x7

#### Example 3

Differentiate g(x) = √x.

#### Solution 3

To use the Power Rule, we need g(x) expressed as x raised to a power, or:

Notice how much easier it is to use the Power Rule to solve this problem than it was using the limit definition of the derivative.

#### Example 4

Differentiate y =

#### Solution 4

Again, we have to rewrite y as x–2 so that it becomes x raised to a power.

#### Example 5

Differentiate y = 3t.

#### Solution 5

Notice that means «take the derivative with respect to variable t." Usually, our variable is x, so the derivative of , but sometimes, we have other variables. If y = f(u), then is the derivative with respect to u, for example.

## The Constant Coefficient Rule

The Constant Coefficient Rule is stated as follows: If a function has a constant multiplied in front, leave it while you take the derivative of the rest. This means that because then the derivative of 5x8 is 5 · (8x7) = 40x7. Just imagine that the constant steps aside and waits while you differentiate the rest.

### The Constant Coefficient Rule

If a function has a constant multiplied in front, leave it while you take the derivative of the rest.

#### Examples

Differentiate the following.

 f(x) = 11x4 y = 10x2 g(x) = 3√x = 3x h(t) = = 4t–6 y = 12x k(u) = = A(r) = π r2

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