Differentiation Concepts and Principles

The video tutorial explains the concept of derivatives, starting with basic rules like the power rule and coefficient handling. It then delves into the first principles of derivatives, illustrating how constants factor out during differentiation. Finally, it covers the derivative of constant functions, emphasizing that their gradient is zero.

What happens to the power of x when finding the derivative of x^n?

• The power is multiplied by the coefficient.

• The power decreases by one.

• The power remains the same.

• The power increases by one.

In the context of first principles, what is the role of the constant 'a' when differentiating a function a*g(x)?

• It is added to the derivative.

• It becomes zero.

• It remains constant and can be factored out.

• It changes with h.

What is the significance of function notation in differentiation?

• It is used to denote constants.

• It represents a function being applied to a variable.

• It indicates multiplication.

• It is irrelevant in differentiation.

What happens to the derivative of a function when multiplied by a constant?

• The constant is factored out and multiplied with the derivative.

• The constant is subtracted from the derivative.

• The constant is added to the derivative.

• The constant becomes zero.

What is the derivative of a constant function like f(x) = 3?

• 3

• 1

• 0

• x

What is the gradient of a horizontal line on a graph?

• 0

• Equal to the y-intercept

• 1

• Undefined

What is the derivative of a function f(x) = 5?

• 5

• 1

• x

• 0

How does the derivative of a sum of functions relate to the derivatives of the individual functions?

• It is the quotient of the individual derivatives.

• It is the product of the individual derivatives.

• It is the difference of the individual derivatives.

• It is the sum of the individual derivatives.

When differentiating 3x^2, what is the resulting expression?

• 9x

• 3x

• x^2

• 6x

In the expression f(x) = ax^n, what is the derivative f'(x)?

• nx^(n-1)

• a^2x^(n-1)

• ax^(n+1)

• anx^(n-1)