Understanding Derivatives and Their Applications

The video tutorial from ganithika.com explains how to find derivatives using the first principle method. It begins with an introduction to derivatives and their geometrical meaning. The tutorial then demonstrates the first principle method for calculating derivatives, focusing on polynomial functions and their derivatives. It also covers the derivatives of trigonometric functions, specifically sine and cosine, using the first principle method. The video provides examples and explanations to help learners understand the concepts and apply them to different types of functions.

What is the geometrical interpretation of the derivative of a function at a point?

• The angle between the curve and the x-axis

• The distance between two points on the curve

• The area under the curve at that point

• The slope of the tangent to the curve at that point

In the first principle method, what happens to the chord as point B approaches point A?

• It becomes a secant

• It becomes a tangent

• It disappears

• It remains a chord

What is the derivative of x^n using the first principle method?

• x^(n-1)

• n * x^(n-1)

• x^n

• n * x^n

How is the derivative of a constant function determined?

• It is equal to the constant itself

• It is one

• It is zero

• It is undefined

What is the derivative of 1/x using the first principle method?

• 1/x^2

• -x^2

• x^2

• -1/x^2

How can the square root of x be expressed for differentiation purposes?

• x^(-1/2)

• x^(1/2)

• x^(1/3)

• x^2

Which of the following is a correct expression for the derivative of a polynomial term 3x^5?

• 15x^4

• 3x^4

• 5x^4

• 15x^5

What is the derivative of x using the first principle method?

• x^2

• 1

• x

• 0

What is the derivative of sin(x) using the first principle method?

• cos(x)

• -sin(x)

• tan(x)

• -cos(x)

What is the derivative of cos(x) using the first principle method?

• sin(x)

• -cos(x)

• tan(x)

• -sin(x)