Chapter 14, Section 14.2: Computing Partial Derivatives Algebraically

derivative of a constant

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edfinity #3: zy (1:08:27 in recording) (!!!)

1. with a thing like 5^x, e^ln(5) = 5 2. rewrite as e^(ln(5))^x

Find zx if z = (1/(2x²ay)) + ((3x⁵abc)/y)

1. assume a, b, and c are constants 2. rewrite zx = del/delx (1/(2x²ay)) + del/delx ((3x⁵abc)/y) 3. note the underlying form 4. solve

what is the chain rule?

1. f'(g(x))g'(x) -- derivative of the outside times the derivative of the inside 2. also d/dx (u^n) = n(u^n-1)(du/dx)

edfinity #8: (55:42 in 14.1/14.2 recording) (!!!)

1. probably always safer to calculate the derivative viewing one variable as a constant then plugging in the point 2.

What is d/dy (f(1,2)) of 5x²y³ + 8xy² - 3x²? (!!!)

1. since it's d/dy, set x = 1 2. plug in x = 1 into the equation and solve 3. take the derivative 4. plug in y 5. probably should take the derivative first then plug in point?

Let f(x, y) = x2 / y + 1 . Find fx(3,2) algebraically (!!!)

1. since its delf/delx, set y = 2 2. set y and take the derivative 3. plug in (3,2)

Find fx + fy for f(x,y) = ln(x.²y.³)

1. split into an easier function ln(x.²) + ln(y.³) 2. simplify: .2ln(x) + .3ln(y) 3. delf/delx = del/delx(.2ln(x) + .3ln(y)) 4. since y is a constant, .3ln(y) becomes 0 5. so fx = .2/x 6. repeat for fy, x is now a constant

If f(x,y) = 5x²y³ + 8xy² - 3x², find a formula for fx(x,y)

1. to calculate fx(x,y) = delf/delx(x,y) = delz/delx (x,y), view y as a CONSTANT 2. take derivative with respect to x 3. you can plug in the point if given to find the derivative at that point

If f(x,y) = 5x²y³ + 8xy² - 3x², find a formula for fy(x,y)

1. to calculate fy(x,y) = delf/dely(x,y) = delz/dely (x,y), view x as a CONSTANT 2. take derivative with respect to y

remember, for the power rule, sqrt(x) and 1/x^2 can be

1. written as x^1/2 2. written as x^-2

chain rule d/dx ln(u) =

1/u(du/dx)

What is d/dx (f(x,2)) of 5x²y³ + 8xy² - 3x²?

40x² + 32x - 3x²

e^(ln(5)) =

5

derivative of ln(x)

= 1/x, defined when x > 0

derivative of ln(|x|)

= 1/x, defined when x ≠ 0

d/dx (C (f)) =

C times (d/dx (f))

derivatives of e^(Cx) =

Ce(^Cx)

what is the product rule?

Dx(f x g)=f x g' + g x f'

this section is about

doing algebra for concepts in 14.1

derivative of e^x

e^x

what is the quotient rule?

g(x)f'(x)-f(x)g'(x)/g(x)^2

ln(ab) =

ln(a) + ln(b)

what is the power rule? d/dx(x^n) =

n(x^n-1)

ln(aⁿ) =

nln(a)

book #13/edfinity #5:

see notes: 14.1 + 14.2 SESSION TL;DR double chain-rule for e^x

derivative of the sum is

the sum of the derivatives

T/F: fx(1,2) is the derivative of the one variable function f(x,2) evaluated at x = 1

true

find fx if f(x,y) = e^x(sin x + y)

use product rule

find zy if z = (2x²y³ - y⁴) / (7xy - 2)

use quotient rule

remember, for the chain rule, sin^2(x) can be

written as (sinx)^2

power rule only applies when

x is in the base