Calculus: Derivative Quiz 2, Implicit Differentiation, Mathematica ContourPlot, Partial Derivatives

Find the derivatives of 1) f(x)=4ln(3x)+x^(e), 2) f(x)=sin(2x)/x, 3) f(x)=x^2*e^(-2x), 4) f(x)=tan(x)+tan^(-1)(x), 5) f(x)=(cos(x))^(-1)=sec(x). Also discuss implicit differentiation, visualization with ContourPlot and Plot3D on Mathematica, and partial derivatives from Multivariable Calculus.

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⏱️TIMESTAMPS⏱️
(0:00) Differentiate f(x)=4ln(3x)+x^(e)
(1:58) Find the derivative of f(x)=sin(2x)/x
(3:32) Find f'(x) when f(x)=x^2*e^(-2x)
(4:46) Differentiate f(x)=tan(x)+tan^(-1)(x)=tan(x)+arctan(x)
(8:06) Find the derivative of f(x)=(cos(x))^(-1)=sec(x)
(9:59) Implicit differentiation example (using function notation)
(13:02) Use ContourPlot on Mathematica to graph the equation
(16:30) Find dy/dx at (x,y)=(1,2) (find f'(1), assuming that the equation defines an implicit function of x near x = 1)
(24:52) Use Plot to graph the tangent line (and use Show in Mathematica)
(26:45) Use linear approximation to approximate another point on the graph when x = 1.1
(29:43) Define a function of two variables (a multivariable function) one of whose level curves matches the original implicitly defined function
(33:27) Use Plot3D to make a surface graph of the function in 3-dimensional space
(39:48) Find first order partial derivatives
(44:36) Find second order partial derivatives, including the mixed partials
(47:08) Clairaut's Theorem guarantees the 2nd order mixed partials are equal when the original function is twice continuously differentiable

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