deriv calculus!
TRANSCRIPT
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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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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!