Introduction to Derivation Rules

Introduction to Derivation Rules By Jason Lam d ƒ´(x)= ƒ(x) __ dx Click me!

DIRECTIONS Confused? Here are some directions! Use this to move to the next slide. Use this to move to the previous slide. Use this to move to the Menu. Use this to move to start of the section.

MENU Constant Functions Power Functions Exponent Functions What would you like to derive? Learn how constant functions behave when derived. Learn how to derive functions that have a variable to any power. These are some very simple rules that do not fall under the above categories. They are very useful and necessary for further progress. Challenge your understanding of the above content with this 10 questions quiz. Learn how to derive functions when the exponent is a variable. Extra Rules Quiz

Constant Functions The below proof shows that for a function f(x) where f(x) is a constant value c, the slope is 0. We can also say f’(x) = 0.

Constant Functions So the derivative of a constant function where c is a constant value is as follows: Slope = 0

Constant Functions For example, given a function such that y = 127, what is dy/dx? The derivative of y Substitute for y Apply constant rule So, the derivative of any constant is zero. This also means that the slope of any constant is zero.

Constant Functions You have now learned how to derive constant functions. Click “Menu” for more options or “Restart” to go over constant functions once more. GOOD JOB!

Power Functions Power functions can be derived using the Power Rule: Where “n” is any real number.

Power Functions Given what is f’(x)? Following the power rule, identify x and n: In this case: x = x n = 6

Power Functions The power rule still applies even if the power is negative. Givenfind? Rewrite the equation so that In this case: x = x n = -1 or

Power Functions The power rule still applies even if the power is a fraction. Given find? We use the same technique from the previous slide. Rewrite the equation so that In this case: x = x n = or

Power Functions You have now learned how to derive constant functions. Click “Menu” for more options or “Restart” to go over power functions once more. EXCELLENT!

Exponential Functions The rate of change of any exponential function is proportional to the function itself. In other words, the slope is proportional to the height. This is true for all exponential functions but for now le us work with the function

Exponential Functions The derivative of the natural exponential function is as follows: S is its own derivative. How cool is that?

Exponential Functions Because of this unique property, the following is true: : : And so on

Exponential Functions can be used with various other functions. For example, given find and using the constant and power rule we get:

Exponential Functions You have now learned how to derive constant functions. Click “Menu” for more options or “Restart” to go over exponential functions once more. FANTASTIC!

Extra Rules Here you will learn the constant multiple rule, sum rule and difference rule. They are all very easy to understand will help you derive many variations to common functions.

Extra Rules Constant Multiple Rulestates that Where C is a constant real number and f(x) is a differentiable equation.

Extra Rules So given find . We can find this using the power and constant multiple rule: Find derivative of the given function. First move the constants to the outside of the derivative. Next you can apply the power rule to the given variable. Finally we have the answer!

Extra Rules The Sum/Difference Rulestates that the derivative of a sum/difference of functions is the sum/difference of the individual derivatives. SO: Where f(x) and g(x) are differentiable functions.

Extra Rules So given find . We can find this using the power and sum/difference rule:

Extra Rules You have now learned the Extra Rules part of this lesson. Click “Menu” for more options or “Restart” to go over the extra rules once more. AWESOME!

Quiz Introduction Use the techniques and rules from this lesson to successfully answer all 10 of the following questions. See if you can get 8 of the 10 correct! QUIZ TIME! GOOD LUCK!

Question 1 Given what is f’(x)? A 18.65 B 186.5 C 0 D x

Great Job! Correct! Using the constant rule We know that f’(x) = 0 if f’(x) is a constant value equal to 186.5

Ooooops! That was incorrect, would you like to try again? HINT: Remember what the constant rule tells us about deriving constant values? Click the X button for no. Click the checkmark for yes.

Question 2 Given what is f’(x)? A 1/3 B 1/30 C 0 D 15

Great Job! Correct! Using the constant rule We know that f’(x) = 0 if f’(x) is a constant value equal to

Ooooops! That was incorrect, would you like to try again? HINT: Remember what the constant rule tells us about deriving constant values? Click the X button for no. Click the checkmark for yes.

Question 3 Differentiate the function A B C 0 D

Great Job! Correct! Using the power rule and constant multiple rule:

Ooooops! That was incorrect, would you like to try again? HINT: Remember what the power rule and the constant multiple rule. Start here: Click the X button for no. Click the checkmark for yes.

Question 4 Given find f’(x) A B 2-t C D 2+t

Great Job! Correct! Using the sum/difference rule and power rule:

Ooooops! That was incorrect, would you like to try again? HINT: Remember what the sum/difference rule states that: Click the X button for no. Click the checkmark for yes.

Question 5 Given find f’(x) A B 0 C D

Great Job! Correct! Using the power and constant multiple rule and power rule:

Ooooops! That was incorrect, would you like to try again? HINT: Combine the power rule and the constant multiple rule. Click the X button for no. Click the checkmark for yes.

Question 6 Given find v’(r) A B C D

Ooooops! That was incorrect, would you like to try again? HINT: Combine the power rule and the constant multiple rule. Also keep in mind that is a constant value. Click the X button for no. Click the checkmark for yes.

Question 7 Given find B’(x) A B C D

Great Job! Correct! Using the sum/difference rule and the properties of an exponential function we get:

Ooooops! That was incorrect, would you like to try again? HINT: Use the sum/difference rule along with the properties of the natural log e. Click the X button for no. Click the checkmark for yes.

Question 8 Given find v’(r) A B C D

Ooooops! That was incorrect, would you like to try again? HINT: Rewrite the original function so that it looks like the one below. Also, keep in mind the power rule: Click the X button for no. Click the checkmark for yes.

Question 9 Given find dy/dx A B C D

Introduction to Derivation Rules