Introduction to Differentiation of Small Positive Integer Powers of X

The video tutorial covers the concept of differentiation from first principles, demonstrating how to find the gradient of a function. It provides examples of differentiating functions like X squared, X cubed, and X to the fifth, using methods such as Pascal's triangle and the binomial theorem. A general formula for differentiation is derived, and its application is shown through various examples.

What is the gradient of a function at a point known as?

• The area under the curve

• The y-intercept

• The x-intercept

• The slope of the tangent

When differentiating X^2, what is the resulting gradient function?

• 3X

• 2X

• X^2

• X

Which theorem is used to expand (X + H)^3 in the differentiation process?

• Pythagorean Theorem

• Binomial Theorem

• Fermat's Last Theorem

• Pascal's Triangle

What is the gradient function for X^5 after differentiation?

• 5X^4

• 5X^3

• 4X^5

• X^5

What is the general formula for the gradient of Y = AX^n?

• nX^(n+1)

• AX^n

• nAX^(n-1)

• AX^(n-1)

In the example Y = X^4 - 5, what is the gradient when X = 3?

• 108

• 81

• 27

• 54

For the function Y = 3X^3 - 6X^2 + 2X - 7, what is the gradient at X = -2?

• 36

• 62

• 24

• 12