Deriv calculus!
- 2. The Table of Derivative Rules •What is a derivative? •Exponent Rule •Product Rule •Chain Rule •Trig. Rules •Natural Log •eˣ
- 3. What’s a Derivative 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!
- 4. Express a Derivative? A derivative can be expressed by: • the lim as x → 0 of dy/dx • (f(x-Δx) – f(x))/ Δx • dy/dx
- 5. The Rules
- 6. 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
- 7. Product rule applies when deriving two functions •This Rule 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’
- 8. 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
- 9. Chain Rule One 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 equation as it may include other rules.
- 10. 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)
- 11. New Rules (ln and ex) • ln(x) the derivative of this is 1/x • ex the derivative of this is ex These are mostly memorization as well, however you’ll be surprised that one of these functions derivatives is the same as the function.
- 12. Now you’re ready… FOR THE QUIZ!