Studied by 6 peopleCalculus
Derivatives & Differentiation
AP Calculus AB
Unit 2: Differentiation: Definition and Fundamental Properties
AB Calc Unit 2
Derivative of sinx
cosx
Derivative of cosx
-sinx
Derivative of tanx
sec²x
Derivative of secx
tanxsecx
Derivative of cscx
-cotxcscx
Derivative of cotx
-csc²x
How do you remember the signs for the trig derivatives?
If it starts with a c, the derivative is negative. If not, stay positive!
Trig reciprocals in a calculator
cscx - 1/sinx
secx - 1/cosx
cotx - 1/tanx
How is a line tangent to a function?
It must be “locally linear”
Limit definitions of instantaneous rate of change (f’(a)) at x=a?
f’(a)= limh→0(f(a+h) - f(a))/h) OR f’(a) = limx→a(f(x)-f(a))/x-a)
Limit definition of the function of the derivative?
f’(x) = limh→0(f(x+h) - f(x))/h
How do you find the slope of the tangent line at (a, f(a))?
Find the slopeusing the derivative and a point: y-f(a)=mtan(x-a)
Symmetric Form
f’(a)=f(c+h)-f(c-h)/2h
If an FRQ asks you to find average velocity, you use…
AROC
What is special about the derivatives of linear functions?
Their derivatives equal the slope of the line!
When it comes to determining wether f(x) is increasing or decreasing at extrema…
it’s doing both, so use a square bracket when including its x-value
Properties of derivatives based on the original function:
extrema → zeroes
where f is increasing/decreasing → x-values on f’ are positive/negative
if f is concave up/down → f’ is increasing/decreasing
where f has a point of inflection → f’ has extrema
Relationship between continuity and differentiability
Differentiability guarantees continuity, but continuity does not guarantee differentiability
A function is not differentiable at…
cusps, corners, discontinuities, endpoints, and points with vertical tangents
Derivative of a constant
0
Power rule
d/dx[xn] = nxn-1
Product rule
lowDhi+hiDlow
Quotient rule
lowDhi-hiDlow/low²
Note
Flashcard