Differentiation

Question 1

Differentiate y=[f(x)]^n

Answer 1

dy/dx=n[f(x)]^(n-1) f’(x)

Question 2

Differentiate y=f[g(x)]

Answer 2

dy/dx=f’[g(x)] g’(x)

Question 3

Differentiate y=uv

Answer 3

dy/dx=uv’ + vu’

Question 4

Differentiate y=(u(x))/(v(x))

Answer 4

dy/dx= (vu’ - uv’) / v^2

Question 5

Differentiate y=e^(f(x))

Answer 5

dy/dx= f’(x) e^(f(x))

Question 6

Differentiate y=ln(x)

Answer 6

dy/dx= 1/x

Question 7

Differentiate y=ln[f(x)]

Answer 7

dy/dx= f’(x) / f(x)

Question 8

Differentiate y=a^x

Answer 8

dy/dx=a^x ln⁡(a)

Question 9

How do you differentiate a parametric equation with n variables? z=f(g(t),h(t))

Answer 9

dz/dtj = f’(x1) . [dx1/dtj] +⋯+f’xn . [dxn/dtj]

Question 10

What is an implicit function?

Answer 10

A function with y and x in z=f(x,y)

Question 11

How do you differentiate an implicit function? z=f(x,y)

Answer 11

1. Differentiate both x + y terms → multiply differentiated y terms by dy/dx (chain rule)
2. Make dy/dx the subject of the equation

f(y)→ f’(y) . dy/dx

Question 12

How do you differentiate the implicit function f(x)∙g(y)?

Answer 12

f(x)∙g(y)→ f(x)∙g’(y)∙dy/dx +g(y)∙f’(x)

Question 13

How do you differentiate the implicit function z=f(x,h(x)) ?

Answer 13

dz/dx=(∂z/∂x)+(∂z/∂y)∙(dy/dx)

→ since dx/dx = 1

Question 14

What is a partial derivative?

Answer 14

A function differentiated with respect to 1 variable

Question 15

What are the notations for partial, and cross partial, deviates.

Answer 15

∂f/x , f’1 , f’x or fx → 1st partial derivative taken with respect to the 1st variable (x)

∂^2f/x^2 , f’‘1 , f’‘xx or fxx → 2nd partial derivative taken with respect to 1st variable 2ce

∂^2f/∂xy , f’‘12 , f’‘y or fxy → cross partial derivative, 2nd argument is the one that is being differentiated 2nd

Question 16

What is a Hessian matrix

Answer 16

H=fxx fxz

fzx fzz

Question 17

State Young’s Theorem

Answer 17

fzx=fxz → if all mth-order partial derivatives are continuous cross-partial derivatives with the same variables are equal

Question 18

When k is the no. of variables in f - how many partial derivates do you need to calculate?

Answer 18

Combining hessian matric + young’s theory → when k is the no. of variables in f you only need to compute K+K*(K-1)/2 partial derivatives

Question 19

What does a total differentials show?

Answer 19

They are a measure of change of a function when many of its arguments change.

Question 20

How do you calculate a total derivative? (for z=f(x,y) and z=f(x,x1,…,xn)

Answer 20

∆z=fx’∆x+fy’ ∆y
or
dz=fx1’.dx1+⋯+fxn’dxn

Question 21

What is the total derivative of d(af+bg)?

Answer 21

d(af+bg)=a∙df+b∙dg

Question 22

What is the total derivative of d(fg)?

Answer 22

d(fg)=g∙df+f∙dg

Question 23

What is the total derivative of d(f/g)?

Answer 23

d(f/g)=(g.df-f.dg)/g^2 g≠0

→ Alternatively think d(fg^(-1) ) and use the product rule

Question 24

What is the total derivative of z=g(f(x,y))?

Answer 24

z=g(f(x,y)) → dz=g^’ (f(x,y))df