deriv calculus!

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The DERIVATIVE RULES!

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•What is a derivative?•Exponent Rule•Product Rule•Chain Rule•Trig. Rules•Natural Log•eˣ

The Table of Derivative Rules

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Derivative is the instant rate of change. There are several rules which make it easier to find the derivative. Today we will discover those rules!

What’s a Derivative

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A derivative can be expressed by:• the lim as x → 0 of dy/dx• (f(x-Δx) – f(x))/ Δx• dy/dx

Express a Derivative?

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The Rules

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Exponent Rule

• The simplest of the derivative rules, applied to a letter such as x to a specific power.

• xⁿ = nxn-1

• Example:• x5 = 5x4

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Product Rule• This rule applies when deriving two functions

which are multiplied.

• f(x)g(x) is two functions multiplied

• the derivative is: g(x)f’(x) + f(x)g’(x)• The process is as follows

• make one function A and the other B; in this case A is f(x) and B is g(x)

• Then derive each function separately• then multiply B*A’ and add it to A*B’

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Quotient Rule• This rule applies when deriving two functions

which are being divided.

• f(x)/g(x) is an example.

• The derivative is (g(x)f’(x) – f(x)g’(x))/((g(x)2)

• the process reach the derivative is as follows

• make one function A and the other B; in this case A is f(x) and B is g(x)

• derive each function separately• then multiply B*A’ and A*B’• then B*A’ – A*B’• then divide by B2

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Chain RuleOne of the most important derivative rules, the chain rule applies when one function is within the other.

For example : f(g(x))

To derive this, by using the chain rule, first derive the f function so in this case it would be f’g(x)

Then derive the inside function in this case it would be g’(x)

Then multiply the derivatives together making a final answer of f’g(x) *g’(x)

Pay close attention to the inside equationas it may include other rules.

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Trig. Rules The easy stuff! Just memorize what the derivative of each function is and if needed apply the other rules.

Function Derivative

Sin(x) Cos(x)

Cos(x) -Sin(x)

Tan(x) Sec2(x)

Cot(x) -CSC2(x)

Sec(x) Sec(x)Tan(x)

CSC(x) -CSC(x)Cot(x)

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New Rules (ln and ex)

These are mostly memorization as well, however you’ll be surprised that one of these functions derivatives is the same as the function.

• ln(x) the derivative of this is 1/x

• ex the derivative of this is ex

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Now you’re ready…

FOR THE QUIZ!