Calculus for Machine Learning and Data Science (3)

Derivatives and Optimization

Derivatives

Existence of the derivative

01

Functions where we cannot find the derivative at every point are called the non-differentiable functions.

At which point in this function does the derivative not exist?

x = 1

Correct. There is a cusp at x = 1 so the derivative does not exist.

At which point in this function does the derivative not exist?

x = -1

Correct. There is a discontinuity/jump at x=-1, so the derivative does not exist.

01

Vertical tangents are not differentiable at the origin (x = 0) because we cannot divide by 0.

Properties of the derivative: Multiplication by scalars

012

When the function is multiplied by a scalar then the derivative is also multiplied by a scalar.

Properties of the derivative: The sum rule

The derivates can be added if they are heading in the same direction.

What is the speed of the child to the earth?

Speed = 0.65 m/s

Correct. The speeds add, to get 0.6 + 0.05 = 0.65

01

If $f(x)=2x$ and $g(x) = x^2$, then what is $(f+g)'(x)=?$

$2 + 2x$

Properties of the derivative: The product rule

012

The products of derivatives are the addition of two separate derivatives multiplied by the other element.

Given that $f(x) = xe^x$, calculate $f'(x)$

$e^x + xe^x$

Properties of the derivative: The chain rule

01

With the above example, the chain rule allows us to find the change in temperature to time if we know the change in height to time and the change in temperature to height.

It allows us to find the derivative of the earlier element if we know the derivatives of the later elements.

If $f(x)=e^{2x}$, then $f'(x)=?$

$2e^{2x}$

Correct! The derivative of $2x$ is $2$ and $f(x)=g(2x)$ where $g(x)=e^x$.

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