Section 1: Derivatives (Rate of Change)

1. Define a derivative and explain it as a rate of change using a real-world example such as speed.
2. For the function f(x) = x², explain what the derivative represents at a given point.

f′(x)=2xf'(x)=2xf′(x)=2x

1. Using the derivative of f(x) = x², calculate the slope at x = 3 and interpret the result.
2. Explain how the slope of a curve differs from the slope of a straight line.
3. For the function f(x) = 3x + 5, explain why the derivative is constant and what it means.

f′(x)=3f'(x)=3f′(x)=3

1. Explain how derivatives help measure how fast something is changing in data.
2. Describe the geometric meaning of a derivative in terms of tangent lines.
3. For the function f(x) = x³, describe how the slope changes as x increases.

Section 2: Gradient (Slope in Multiple Dimensions)

1. Define gradient and explain how it extends the idea of derivative to multiple variables.
2. For a function z = x² + y², explain what the gradient represents at a point.

∇f=(2x,2y)\nabla f = (2x, 2y)∇f=(2x,2y)

1. Explain why the gradient points in the direction of steepest increase.
2. Describe how gradient is used to understand changes in machine learning models.
3. Compare slope in one dimension with gradient in two dimensions.
4. Explain the meaning of partial derivatives using a simple example.