Differentiation Flashcards
Differentiate y=[f(x)]^n
dy/dx=n[f(x)]^(n-1) f’(x)
Differentiate y=f[g(x)]
dy/dx=f’[g(x)] g’(x)
Differentiate y=uv
dy/dx=uv’ + vu’
Differentiate y=(u(x))/(v(x))
dy/dx= (vu’ - uv’) / v^2
Differentiate y=e^(f(x))
dy/dx= f’(x) e^(f(x))
Differentiate y=ln(x)
dy/dx= 1/x
Differentiate y=ln[f(x)]
dy/dx= f’(x) / f(x)
Differentiate y=a^x
dy/dx=a^x ln(a)
How do you differentiate a parametric equation with n variables? z=f(g(t),h(t))
dz/dtj = f’(x1) . [dx1/dtj] +⋯+f’xn . [dxn/dtj]
What is an implicit function?
A function with y and x in z=f(x,y)
How do you differentiate an implicit function? z=f(x,y)
- Differentiate both x + y terms → multiply differentiated y terms by dy/dx (chain rule)
- Make dy/dx the subject of the equation
f(y)→ f’(y) . dy/dx
How do you differentiate the implicit function f(x)∙g(y)?
f(x)∙g(y)→ f(x)∙g’(y)∙dy/dx +g(y)∙f’(x)
How do you differentiate the implicit function z=f(x,h(x)) ?
dz/dx=(∂z/∂x)+(∂z/∂y)∙(dy/dx)
→ since dx/dx = 1
What is a partial derivative?
A function differentiated with respect to 1 variable
What are the notations for partial, and cross partial, deviates.
∂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
What is a Hessian matrix
H=fxx fxz
fzx fzz
State Young’s Theorem
fzx=fxz → if all mth-order partial derivatives are continuous cross-partial derivatives with the same variables are equal
When k is the no. of variables in f - how many partial derivates do you need to calculate?
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
What does a total differentials show?
They are a measure of change of a function when many of its arguments change.
How do you calculate a total derivative? (for z=f(x,y) and z=f(x,x1,…,xn)
∆z=fx’∆x+fy’ ∆y
or
dz=fx1’.dx1+⋯+fxn’dxn
What is the total derivative of d(af+bg)?
d(af+bg)=a∙df+b∙dg
What is the total derivative of d(fg)?
d(fg)=g∙df+f∙dg
What is the total derivative of d(f/g)?
d(f/g)=(g.df-f.dg)/g^2 g≠0
→ Alternatively think d(fg^(-1) ) and use the product rule
What is the total derivative of z=g(f(x,y))?
z=g(f(x,y)) → dz=g^’ (f(x,y))df
