Understanding Derivatives: From Basic Concepts to Tangent Lines and Applications in Deep Learning

Where is derivatives used in neural network?

The Blue Curve (f(x)):

• This represents the original function, let's call it f(x).
• In this case, it's a parabola shape, similar to y = x².

The Red Line (f'(x)):

• This represents the derivative of the function, f'(x).
• It shows how the rate of change of the original function varies.

Derivatives play crucial role in deep learning, particularly while training neural networks.

1. Gradient Descent:
2. Deep learning models are trained using an optimization algorithm called gradient descent.
3. This algorithm uses derivatives to find the minimum of the loss function, which represents how well the model is performing.
4. Back propagation:
5. This is the primary algorithm for training neural networks.
6. This uses chain rule of derivatives to efficiently compute gradients for all parameter in the network.
7. This allows error signal to propagate backwards through the network, adjusting weights to minimize the loss.
8. Activation functions:
9. Non-linear activation in neural networks often have specific derivative properties.
10. The popular Relu (rectified linear unit) function has a simple derivate: 1 for positive inputs and 0 for negative.
11. The ease of computing derivatives contributes to faster training.
12. Learning Rate:
13. The learning rate in deep learning algorithms is essentially a scaling factor for the derivative.
14. Which determines how large of a step to take in the direction of the negative gradient
15. Feature importance:
16. The derivative of the output with respect to the input feature can indicate how important that feature is for the model's prediction.

In conclusion, derivatives are the mathematical backbone of how deep learning models learn. They convey necessary information about how to adjust the model's parameters to improve it performance.

Understanding derivatives is crucial for developing new architectures, optimizing existing ones, solving issues in deep learning models.

What is Derivative?

Derivative is a fundamental concept in calculus that measures how function changes as its input changes.

Basic definition: The derivative of a function at a given point is the instantaneous rate of change of the function at that point.

Rate of change: The derivative tells us how quickly the function's output is changing with respect to its input at any specific point.

Sign interpretation:

• A positive derivative means the function is increasing.
• A negative derivative means the function is decreasing.
• A zero derivative indicates a flat point or local extremum (minimum or maximum).

The derivative at red point is the slope of that red tangent line. This slope tells us how fast the car's position is changing at that exact moment - its instantaneous speed.

• The derivative would be written as f'(t) or df/dt
• This f'(t) gives us the car's speed at any moment t.

To start with a simple example. Let's start with the idea of rate of change.

1. Imagine you are driving a car, your speed is rate of change
2. it tells you how quickly your position is changing over time.
3. If you are going 70 miles/hour, that means in 1 hour you'll travel 70 miles. This is an average rate of change.
4. But what if we want to know your speed at a specific moment?
5. we want to know your exact speed at a precise moment (4:35:33 PM)
6. thats where derivatives come in, they give us the instantaneous rate of change.

Here's what the derivative (instantaneous rate of change) tells us:

1. At exactly 4:35:33 PM, your instantaneous speed is represented by the slope of that red tangent line.
2. This slope gives us your speed at that precise moment - not your average speed over the whole trip.

Understanding the Scenario:

Accelerating to 70 mph:

1. From 4:00 PM to 4:30 PM, you're increasing your speed (positive acceleration).
2. This is represented by the positive slope on the graph.
3. Your velocity is increasing, meaning you're gaining speed, like accelerating in a car.

1. Reaching the Peak Speed (70 mph):At some point, you reach your maximum speed, let’s say 70 mph (this happens around 4:30 PM).
2. The slope of the graph at this point is zero because you're neither accelerating nor decelerating at that moment (like cruising at a constant speed).

Slowing Down After 4:30 PM:

1. After 4:30 PM, you start slowing down—your speed decreases (69 mph, 68 mph)
2. On the graph, this is shown as the negative slope. You're still moving forward, but your velocity is decreasing. You're decelerating.

So in this analogy, the positive and negative slopes of the graph represent how your speed is changing:

• Positive slope: Accelerating.
• Negative slope: Decelerating.

But before we understand the rate of change, we need to understand

Tangent:

Imagine you're on a rollercoaster ride.

The roller coaster track represents our function - it goes up, down, and curves around.

1. Your position:
2. As you ride the roller coaster your position is constantly changing. At any given moment, you're at a specific point on the track.
3. Your speed:
4. At any instant, you're moving in a particular direction at a certain speed
5. The Tangent:
6. A line that represents the direction you're moving at a specific moment.
7. If you suddenly fly off the track, you'd move along this straight line.
8. And the line is called Tangent.

The term tangent in mathematics captures the essence of a line that just touches a curve at a single point, providing crucial information about the curve's behaviour at that specific location. This concept is fundamental in calculus for understanding instantaneous rate of change & behaviour of functions.

The x-axis represents distance (horizontal movement).

The y-axis represents height (vertical movement).

What behaviour can be concluded from these 3 different tangent?

1. Tangent at Point A (red):
2. Located at the top of a hill
3. The tangent line has a steep negative slope
4. This indicates the roller coaster is about to enter a sharp descent
5. Riders would experience a sudden drop and acceleration downwards
6. if the roller coaster track disappeared, you'd fall, face straight on ground in the direction represented by the tangent.
7. Tangent at Point B (green):
8. Located at the bottom of a dip
9. The tangent line is nearly horizontal with a slight negative slope
10. This represents the lowest point of the track where the coaster transitions from descending to ascending
11. if the track disappeared, you'd slowly move on direction on the right represented by tangent
12. Tangent at Point C (purple):
13. Located on an upward slope
14. Slope: Moderate positive slope. You're climbing the hill.
15. Positive vertical acceleration (riders feel pushed into their seats)
16. If the track disappeared, you'd continue upward and forward along this line.

Slope, Tangent & Derivate:

Slope:

• It's calculated as rise (vertical change) divided by run (horizontal change).
• The slope is how steep the tangent line is.

Function (Blue Curve):

• This is the original function, let's call it f(x).
• It represents how y changes as x changes.

Tangent Line (Red Line):

• This is a straight line that touches the curve at a single point (Point P).
• It represents the direction of the curve at that specific point.

Derivative:

• Not shown in the image, but it's closely related to the tangent and slope.
• At any point on the curve, the derivative f'(x) equals the slope of the tangent line at that point.

To make sense:

• The function tells us "where we are"
• The tangent line shows "which way we're going" at a specific point
• The slope tells us "how steep our path is" at that point
• The derivative gives us a formula to calculate this steepness for any point on the curve

Derivate will only tell us, how steep our path is for any given point in the curve?

The derivative provides more information than just steepness:

• Positive derivative: function is increasing
• Negative derivative: function is decreasing
• Zero derivative: function is neither increasing nor decreasing (local max, min, or inflection point)
• Larger absolute value: faster rate of change
• Smaller absolute value: slower rate of change

Does it tell anything about tangent?

Derivative tells us something very important about tangent line.

1. The derivative at a point is equal to the slope of the tangent line at that point.
2. When we calculate a derivative, we're finding a function that gives us the slope of the tangent line at every point

Visualizing the relationship:

• Imagine "zooming in" on any point of the curve
• As you zoom in closer and closer, the curve starts to look more and more like a straight line
• This straight line is the tangent line, and its slope is given by the derivative at that point

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