My (Chiffon) Nguyen

differentiation

Feb 25, 20252 min read

Single Variable Differentiation

Common derivatives

Common derivatives

Constant, Exponential and Logarithmic Functions

Trigonometric Functions

Inverse Trigonometric Functions

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Differentiability

differentiable

function is differentiable at if its has a derivative there. Formally, the following two-sided limit expression must exist:

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Rules

Rules

Algebra of derivatives

  1. Multiply a function by a constant
  1. Add or subtract two functions

power rule

chain rule

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Applications

(Khan Academy)

Find rate of change

You can't use 'macro parameter character #' in math mode\begin{gather} v=\frac{dx}{dt} \\ a=\frac{dv}{dt}=\frac{d^{{2}x}}{dt^{2}} \end{gather}$$ $f'(t)$ is the *instantaneous rate of change of $f(x)$ at $x=t$* () ### Differentiate related functions _Differentiate_ equations that relate multiple [[function|functions]] of the same variable > [!example]- Example > > The height $h(t)$ of a rectangle is increasing at a rate of 11 cm per hour and the width $w(t)$ of the rectangle is decreasing at a rate of 9 cm per hour. At a certain instant $t_0$, $h(t_{0})=3cm$ and $w(t_{0})=$. What is the rate of change of the area $A(t)$ of the rectangle at that instant? ### Approximation ![[approximation]] ### Related rates ![[related rates]] ### Find limits with L'Hopital's rule ![[L'Hopital's rule]] ### Optimization ![[mathematical optimization#process|Process]] ### Critical points (i.e. local extrema) ![[critical point#single-variable-calculus|Single Variable Calculus]] ### Absolute extrema ![[global extremum]] ### Derivative tests | | f'>0 | f'<0 | | ----- | -------------------------- | ---------------------------- | | f''>0 | f $\uparrow$, concave up | f $\downarrow$, concave up | | f''<0 | f $\uparrow$, concave down | f $\downarrow$, concave down | ![[second derivative test]] ### Inflection point ![[inflection point]] # Multivariable Differentiation ## Implicit differentiation ![[implicit differentiation]] ## Partial derivative ![[partial derivative]]

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