Chapter 3- Differentiation

Question 0

Differentiable

Answer 0

If limit exists

We say that f is differentiable at x = a

Question 1

Derivative of f(a)

Answer 1

f(a + h) - f(a) f(x) - f(a)
lim —————– = lim ————–
h->0 h x-> a x - a

Question 2

Tangent line in point-slope form

Answer 2

y - f(a) = f’(a) (x-a)

Question 3

To calculate f’(a) using the limit definition

Answer 3

1. Write out the numerator of the difference quotient
2. Divide by h and simplify
3. Compute the derivative by taking the limit

Question 4

Power rule

Answer 4

d
— x^n = nx^(n-1)
dx

Question 5

Sum rule

Answer 5

(f + g)’ = f’ + g’

Question 6

Constant multiple rule

Answer 6

(cf)’ = cf’

Question 7

d
— e^x
dx

Answer 7

e^x

Question 8

Differentiability implies continuity

Answer 8

However, there exist continuous functions that aren’t differentiable