Differentiation

Question 1

How to show a function is differentiable:

Answer 1

At point x_0 through differentiation from first principles show that as h approaches 0 from both sides of x_0 that they are equal

Question 2

Differentiation from first principles:

Answer 2

Our gradient equation m =y2-y1/x2-x1, replace with our function and +h and find limit when h approaches 0

Question 3

Are all differentiable functions continuous?

Answer 3

Yes

Question 4

Does continuity imply differentiability?

Answer 4

No, as a circle is continuous but not differentiable for example as it in not one to one

Question 5

Product rule

Answer 5

d/(d(x) (u(x)v(x)) = u’(x)v(x) + u(x)v’(x)

Question 6

Quotient rule

Answer 6

d/(d(x) (u(x)/v(x)) = (u’(x)v(x) -u(x)v’(x))/(v(x))^2

Question 7

Chain rule

Answer 7

d(dx) f(g(x)) = f’(g(x)) g’(x)

Question 8

Implicit differentiation

Answer 8

Treat y as a function of x so differentiate as you would x but y values will be multiplied by dy/dx then we rearrange for dy/dx to solve

Question 9

Parametric differentiation

Answer 9

Combination of two functions with a common variable e.g. y(t) and x(t)
Find dy/dt and dx/dt
to find dy/dx use:
dy/dx = dy/dt *dt/dx
for 2nd order = d/dt(dy/dx) *(dt/dx)

Question 10

Leibniz formula used when?

Answer 10

We have an need to calculate a high order derivative with product rule

Question 11

Leibniz formula

Answer 11

Like binomial theorem except instead of powers it is derivatives of function

Question 12

How to determine if a function is differentiable

Answer 12

Use differentiation by first principles.