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





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





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?





The Rules





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





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’





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





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.





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)





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





Now you’re ready…

FOR THE QUIZ!




deriv calculus!

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