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Basic Differentiation Rules
Lesson objectives Teachers' notes
College Board® Topics:Topic 2.5: Applying the Power RuleTopic 2.6: Derivative Rules: Constant, Sum, Difference, and Constant Multiple Topic 2.7: Derivatives of cos x, sin x, ex, and ln x
College Board® Learning Objectives:FUN3: Recognizing opportunities to apply derivative rules can simplify differentiation. FUN3.A: Calculate derivatives of familiar functions. LIM3.A: Interpret a limit as a definition of a derivative.
Teachers' notesLesson objectives
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Lesson notes:
Find derivatives using basic rules of differentiation for polynomial, power, sine, cosine, tangent, exponential and logarithmic functions
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The derivative of a constant function is 0. For any real number, c :
1. The Constant Rule
BASIC DIFFERENTIATION RULES
The slope of a horizontal line is 0.
The derivative of a constant function is 0.
x
y
A.) B.) C.) D.)
2. The Single Variable Rule:
The derivative of x is 1.
Think graphically about the line y = x.
A.) B.) C.)
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3. The Power Rule
If n is a rational number then the function is differentiable and
EX #3: Use the power rule to find the derivatives of the following, if:
A.) B.)
C.) D.)
E.) F.)
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4. The General Power Rule:
real numbers, is, where a and n areThe derivative of the term
STEPS:
1. Multiply the coefficient by the variable's exponent. If no coefficient is stated in other words, the coefficient equals 1 the exponent becomes the new coefficient.2. Subtract 1 from the exponent.
A.) B.)
C.) D.)
EX #4: Use the general power rule to find the derivatives of the following, if:
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E.) F.)
5. The Sum and Difference Rules:
The derivative of a sum or difference is the sum or difference of the derivatives.
and
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A.)
B.)
C.)
D.)
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E.)
F.)
6. Derivatives of Sine, Cosine, and Tangent Functions:
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EX #4: The derivative of the sine function is the cosine function. Consider where the sine function increases, the cosine function is positive and vice versa!
Graphing Derivatives with a Calculator
You will enter the function as a function of x, to be evaluated at the value of x, instead of an actual number value.
TI‐83+ or TI‐84+
Use the numeric derivative operation
(MATH > 8) in the graph editor screen.
TI‐Nspire CX
Use the template menu in the graph editor screen
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EX #5: Using the limit definition for the derivative
we can derive for each of the following trigonometric functions. You will verify the derivative of the sine and cosine functions by using the sum and difference formulas that we learned in PreCalculus.
EX #5: Find the derivative of the sine function by using the limit process.
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EX #5: Find the derivative of the cosine function by using the limit process.
7. Derivatives of the Natural Logarithmic and Exponential Functions:
The derivative of is the only function that has itself as its derivative.
If then
Also,If then , for
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EX #6: Use the laws of logarithms to find the derivatives of the following:
A) B)
C) D)
E) F)