Calculus and Derivatives

Calculus and Derivatives Emily Halsmer

Derivative Rule Lesson In the past, we have learned what the derivative means, and what the derivative of constants, scalars, sums, and differences are. We’ve also had countless years of algebra. So use these skills to make “taking derivatives easier.” You know that derivatives give you the velocity and acceleration of a partial at a specific point on the position function. With these equations derived, you can even find the velocity reacted by “The Beast” our favorite roller coaster ride!! That way you can know exactly how fast you are going at any given time you ride it next time! Enjoy learning the rules to taking some specific derivatives in functions. They would be too complicated with the information that we know currently, but not for long!

Derivative Rule Lessons • Power Rule • Produce Rule • Quotient Rule **Derivative Rule Test**

Power Rule • If you have a function… • in terms of “u” • you are taking the derivative in terms of x • “u” is to some power • Move the exponent to the front of the equation and multiply by the constant. Then subtract one from the original exponent. Home Example 1 Example 2 Try on Own

Power Rule • Example 1: Find the first derivative in terms of x • Solution: Home Power Rule Lesson Example 2 Try on Own

Power Rule • Example 2: Find for • Solution: Home Power Rule Lesson Try on Own

Power Rule • 1.) Find f ’(x) and choose the correct answer. • . • . • . Home Power Rule Lesson

Power Rule • Answer is incorrect… let’s see how we should look at this… • Solution: (5 and 4 are both exponents) (2 and 1 are constants) Take and separately So Try Again

Your Answer is Correct! Way to Go You Rock  Home

Product Rule To take the derivative… • Treat as two different functions and multiply one’s derivative by the others function and add the two • Remember this little saying… The first times the derivative of the second, plus the second times the derivative of the first… this is all there is to this rule Home Example 1 Example 2 Try on Own

Product Rule • Example1: Find f‘(x) when **treat with u and v (both by the power rule) Product Rule says: so… and simplify to Home Product Rule Lesson Try on Own

Product Rule • Example 2: Find f ‘(x) when **in this case By using Product Rule: (then simplify) Home Product Rule Lesson Try on Own

Product Rule • 1.) Find f’(x) and choose the correct answer. • . • . • . Home Product Rule Lesson

Product Rule • Answer is incorrect… let’s see how we should look at this problem… • Hint: 3 is just a constant so leave out to end or multiply into problem! Try Again

Quotient Rule • Just remember this little saying… • Low d’(high) minus high d’(low), all over low squared • In other words… it is the denominator of the function, times the derivative of the numerator, minus the numerator times the derivative of the denominator. This is all over the denominator squared • The “high” in the saying is the numerator (u) • The “low” in the saying is the denominator (v) • The “d” is the derivative Home Example 1 Example 2 Try on Own

Quotient Rule • Example 1: Find f’(x) when • Can simplify with algebra then to finish Home Quotient Rule Lesson Example 2 Try on Own

Quotient rule • Example 2: Find f’(x) when Home Quotient Rule Lesson Try on Own

Quotient Rule • 1.) Find f’(x) • . • . • . Home Quotient Rule Lesson

Quotient Rule • Answer is incorrect… let’s see how we should look at this problem… Try Again

**Derivative Rule Test** • 1.) Which Rule would you use for finding the derivative of • Product Rule Only • Power Rule Only • Quotient Rule Only • Product and Power Rules • Quotient and Power Rules • Product and Quotient Rules

**Derivative Rule Test** • Your answer is incorrect. • The correct answer is the Product AND Power rule! • With the power rule just treat 5 as a constant after you distribute it into the problem. • With the product rule and ,you will get the same thing as using the product rule because the derivative of a constant is zero. Try Again

Your Answer is Correct! Way to Go You Rock  Whooo-hooo!! Next

**Derivative Rule Test** • 2.) Find f’(x) • . • . • . • . • . • .

**Derivative Rule Test** • Your answer is incorrect. • The correct answer is • By using the product rule Try Again

**Derivative Rule Test** • 3.) Find f’(x) • . • . • . • . • . • .

**Derivative Rule Test** • Your answer is incorrect. • The correct answer is • This is by using the power rule • Hint: The powers are 2 and 3; constants are 2,3, and 5. Try Again

**Derivative Rule Test** • 4.) Find f’(x) • . • . • .

**Derivative Rule Test** • Your answer is incorrect. • The correct answer is • You get this by doing the quotient rule Try Again

**Derivative Rule Test** • 5.) Find f’(x) Set up but do now evaluate. • . • . • . • .

**Derivative Rule Test** • Your answer is incorrect. • The correct answer is • This problem is harder because you need to do the product rule as well as the quotient rule. • Use the product rule for finding u’ • Then use the quotient rule for the overall function and you will receive this answer. Try Again

!You Are Finished!  You have now completed the entire section for the power, product, and quotient rules! You have also finished your test and review of the lessons. Congratulations! You are now done! If working in the classroom, let your teacher know, and if at home, you can try to work some similar problems out in the book.

Calculus and Derivatives