AP Calculus BC Unit 2

What is the limit definition of a derivative as h →0?

What is the limit definition of a derivative as x → a?

What is the derivative of sin(x)?

cos(x)

What is the derivative of cos(x)?

-sin(x)

What is the derivative of tan(x)? 

sec²(x) 

What is the derivative of cot(x)?

-csc²x

What is the derivative of csc(x)?

-csc(x) cot(x)

What is the derivative of sec(x)?

sex(x) tan(x)

What is the derivative of c is c is any real number? 

0 

What is the derivative of mx?

m

What is the derivative of xⁿ?

n(x^n-1)

What is the derivative of cf(x)?

cf’(x)

What is the derivative of f(x) ± g(x)? 

f’(x) + g’(x) 

What is the derivative of f(x)g(x)?

f’(x)g(x) + f(x)g’(x)

What is the derivative of f(x)/g(x)?

(where g(x) does not equal 0)

f’(x)g(x)-g’(x)f(x) all divided by g(x)²

What is the derivative of f(g(x))?

f’(g(x)) g’(x)

What does it mean if a function is differentiable at a point, m?

A function is differentiable at a point m only if f’(m) exists

If a function is differentiable at a point, m, is it continuous at that point as well? 

Yes, if a function is differentiable at a certain point it must be continuous at that point

If a function is differentiable completely, what does this mean?

if it is differentiable at every point of x in its domain

When are polynomial differentiable?

they are differentiable at all values of x

When are exponential functions differentiable?

they are differentiable at every value of x

When are logarithmic functions differentiable? 

all values where x > 0 

How to check for smoothness?

slopes should approach the same value on both sides

How to check for continuity?

• the function must be defined at x=a

• the limit as x → a must exist

• the limit as x → a must be equal to f(a)