Sec2Sess32

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Unformatted notes below. Notes on my shorthand method

Chain Rule Derivatives and Directional Derivatives More tools to study derivatives

Differentials Implicit Differentiation

y = f(x) dy = f'(x)dx

Example: y=sin^-1(x) x=sin(y) -> dx = cos(y)dy dy/dx = 1/cos(y) = 1/sqrt(1-x^2)

Total Differential All the variables that cause change f(x,y,z) df = fx*dx+fy*dy+fz*dz = pd(x)*d(x)+pd(y)*d(y)+pd(z)*d(z)

Important: the total differential is the sum of the partial derivatives.

Important: d(f) is not the same as change_f d(f) can be considered a "place holder" for a later value d(f) can: 1. encode how change in x,y,z affect f. 2. placeholder for small variations change_x, change_y, change_z to get approximation formula change_f =~ fx*change_x+fy*change_y+fz*change_z 3. divide by something like d(t) to get a rate of change when x=x(t), y=y(t), z=z(t) Chain Rule = d(f)/d(t) = f(x)*d(x)/d(t)+f(y)*d(y)/d(t)+f(z)*d(z)/d(t) (one instance of the chain rule)

Why is this valid?

1st attempt at validating

df = fxdx+fydy+fzdz dx=x'(t)dt, dy=y'(t)dt, dz=z'(t)dt so df = fx*x'(t)dt+fy*y'(t)dt+fz*z'(t)dt divide by dt => get chain rule.

Better validation

change_f ~= fx*change_x+fy*change_y+fz*change_z (in time change_t) change_f/change_t ~= (fx*change_x+fy*change_y+fz*change_z)change_t

when change_t -> 0: change_f/change_t -> d(f)/d(t) , change_x/change_t -> d(x)/d(t) , ...

so d(f)/d(t) = f(x)*d(x)/d(t)+f(y)*d(y)/d(t)+f(z)*d(z)/d(t) ~= becomes = in limit change_t->0

example problems: https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-b-chain-rule-gradient-and-directional-derivatives/session-32-total-differentials-and-the-chain-rule/MIT18_02SC_we_15_comb.pdf