Unit 2 - Derivatives Test Notecards

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)=m_tan(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] = nx^n-1

Product rule

lowDhi+hiDlow

Quotient rule

lowDhi-hiDlow/low²