Differentiation from First Principles

This video tutorial explains differentiation from first principles, covering five examples: x^2, 5x^2, 2x^2 + 3, x^3, and 1/x. Each example is broken down step-by-step, demonstrating how to derive the formula for the derivative using the concept of limits. The tutorial emphasizes understanding the process of finding the gradient at a specific point on a curve and applying the formula consistently across different functions.

What is the purpose of differentiation from first principles?

• To calculate the gradient at a specific point

• To find the area under a curve

• To determine the maximum value of a function

• To solve differential equations

In the context of differentiation from first principles, what does 'h' represent?

• The change in x

• The change in y

• The slope of the tangent

• The function value

What is the formula used for differentiation from first principles?

• (f(x+h) + f(x)) / h

• (f(x+h) - f(x)) / h

• (f(x) - f(x-h)) / h

• (f(x) + f(x-h)) / h

When applying the differentiation formula to x^2, what is the simplified form of (x+h)^2 - x^2?

• x^2 + h^2

• 2xh + h^2

• x^2 + 2xh + h^2

• 2x + h

What is the derivative of 5x^2 using first principles?

• 15x

• 20x

• 10x

• 5x

What is the first step in applying the differentiation formula to 2x^2 + 3?

• Expand (x+h)^2

• Subtract 2x^2 + 3

• Multiply by 2

• Divide by h

What is the derivative of x^3 using first principles?

• 4x^2

• 5x^2

• 3x^2

• 2x^2

What is the simplified form of (x+h)^3 - x^3?

• x^3 + 3x^2 + 3xh + h^2

• 3x^2 + 3xh + h^2

• x^3 + 3x^2h + 3xh^2 + h^3

• 3x^2h + 3xh^2 + h^3

What is the derivative of 1/x using first principles?

• -1/x^2

• -1/x

• 1/x

• 1/x^2

What is the common denominator used when simplifying 1/(x+h) - 1/x?

• x^2 + xh

• x + h

• x(x+h)

• x^2