AP Calculus BC Unit 2 Differentiation-Definition and Fundamental Properties

University:
Ohio State University
Course:
MATH 1151 | Calculus I
Academic year:

2021
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414

Pages:

4
Author:

umfikie3qn

Unit 2: Differentiation - Definition and Fundamental Properties ● Differentiation: Finding the derivative, or rate of change, of a function ○ Average rate of change : The average rate of change of one quantity changing in respect to another ○ Instantaneous rate of change : The rate of change at a specific moment/point ○ Derivative: In simple terms, the derivative of a function is the slope of the secant line of that function, at a certain ‘x’ value. ■ Notation: f’(x), dy/dx, d/dx ■ Secant line: a line between two points ● Equation of a secant line (AKA difference quotient) : ○ Notice how this is basically the slope formula, except that 𝑥1 and 𝑥2 are replaced with x and h. ‘H’ just refers to the difference in the two ‘x’ values ■ Tangent line: a line that touches ONE ‘x’ point on the curve ● To find the derivative of the tangent line, find the equation of the line that’s tangent to the curve at the specified point ○ Definition of Derivative: ■ Ex. Find the derivative of f(x)=3x at x=2 ● Lim h→ 0 [3(x+h)-3(x)]/h = [3x+3h-3x]/h = 3h/h = 3 ○ Lim h→ 0 of 3 = 3 ○ Thus, f’(2) = 3 ○ Rules for a function to be differentiable: ■ A function HAS to be continuous ■ It CANNOT have a sharp corner (absolute value graphs are an example) ■ If the tangent line at a point in the function is VERTICAL, then it’s NOT differentiable at that point ● Power Rule: ○ Much faster way to determine the derivative of a function ○ Ex. Find the derivative of f(x) = 3𝑥 2 , at x=2 ■ according to the power rule, n=2, so f’(x)= 3(2)𝑥 2−1 which equals 6x. ● Then, plug x=2 into 6x, which = 12. Thus, f’(2)=12 ○ *the derivative of a constant is ALWAYS 0, because n=0 for all constants (it’s implied) ● Addition/Subtraction Rule for Derivatives: ○ To find the derivative of a function that consists of adding/subtracting two functions, find the derivatives of each function separately, and then combine them. ■ Ex. Find the derivative of 3𝑥 2 + 2𝑥 8 . ● f’(x) of 3𝑥 2 = 6x ; g’(x) of 2𝑥 8 = 16𝑥 7 ● Thus, the derivative of 3𝑥 2 + 2𝑥 8 is 6x + 16𝑥 7 ● You can keep on finding the second, third, fourth, etc. derivatives of a function too. ○ All you do is keep finding the derivative of that function ■ Ex. The first derivative of 2𝑥 3 = 6𝑥 2 ● To find the second derivative, denoted as f’’(x), just take the derivative of the first derivative, 6𝑥 2 , which equals 12x ● Continue to do this for the third derivative, etc. ● Product Rule: ■ Ex. Find the derivative of h(x) = ( 2𝑥 3 )(4𝑥 2 -3x) ● Let’s set f(x)= 2𝑥 3 and g(x)= 4𝑥 2 -3x ○ h’(x) = (2𝑥 3 )(8x-3) + (4𝑥 2 -3x)(6𝑥 2 ) ● Quotient Rule: ○ Remember: “[(low)(derivative of high) - (high)(derivative of low)]/𝑙𝑜𝑤 2 ” ■ *High refers to the numerator, and low refers to denominator ■ Ex. Find derivative of f(x)=5𝑥 2 /(4𝑥 2 -3x) ● f’(x)= [(4𝑥 2 -3x)(10x)-(5𝑥 2 )(8x-3)] / (4𝑥 2 − 3𝑥)2 ● Derivative of Trig and Inverse Trig functions (it’s best to just memorize all of these): ● Derivative of ln(x) and 𝑒 𝑥 (again, i’d just memorize them for convenience) : ○ f(x) = lnx → f’(x) = 1/x ○ f(x) = 𝑒 𝑥 → f’(x) = 𝑒 𝑥