Calculus for Machine Learning and Data Science (2)

Derivatives and Optimization

Derivatives

Some common derivatives - Lines

Here are some of the common derivatives

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Derivative of a constant and a line

Some common derivatives - Quadratics

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Using simple and formal calculations

For the above slide, plug in the values for $\Delta x$ (the change in $x$) and $x$, where $x = 1$ to get $\Delta f$ and then the slope

Some common derivatives - Higher degree polynomials

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Using simple and formal calculations

Some common derivatives - Other power functions

012

For all power functions, the exponent goes to a scalar value, and the exponent is subtracted by 1 as the new exponent.

The inverse function and its derivative

The inverse function undoes what a function did to an input.

01234

The derivative of the inverse function is 1 over the derivative of the function.

Derivative of trigonometric functions

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As $\Delta x$ gets smaller, the orange triangle also gets smaller, and the hypotenuse $h$ changes to $\Delta x$.

So update the formula and divide both left and right by $\Delta x$, where the left term changes to the derivatives and the right term cancels out the $\Delta x$ that leaves the trigonometric functions.

For formal math explanation, refer to [Derivatives of the Trigonometric Functions](https://www.mathsisfun.com/calculus/derivatives-trig-proof.html#:~:text=The%20three%20most%20useful%20derivatives%20in%20trigonometry%20are...,sin%20%28x%29...%20ddx%20tan%20%28x%29%20%3D%20sec2%20%28x%29).

Meaning of the Exponential (e)

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Using the bank interests as an example.

As n increases the output gets close to the value of e.

The derivative of $e^x​$

The derivative when x = 2

The derivative of $e^x$ is itself.

The derivative of log(x)

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The derivative of a log is the inverse of the value

 

All the information provided is based on the Calculus for Machine Learning and Data Science |  Coursera from DeepLearning.AI