Gr12 Mathematics: Ch 5.2 Differentiation from first Principles

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

f'(x) = [f(x + h) + f(x)]/h

f'(x) = [f(x + h) - f(x)]/h

f'(x) = lim(h → ∞) [f(x + h) - f(x)]/h

f'(x) = lim(h → 0) [f(x + h) - f(x)]/h (correct)

What is the purpose of differentiation from first principles?

To find the maximum value of a function

To determine the gradient of a tangent to a curve (correct)

To find the intersection points of two curves

To find the area under a curve

What is the meaning of the notation dy/dx?

The quotient of y and x

y differentiated with respect to x (correct)

The product of y and x

x differentiated with respect to y

What is the name given to the symbols D and d/dx?

Differential operators Signup and view all the answers

What is the formula for the gradient of a tangent to a curve at a point x = a?

lim(h → 0) [f(a + h) - f(a)]/h Signup and view all the answers

What is the common notation used to denote the derivative of a function f(x)?

f'(x) = y' = dy/dx Signup and view all the answers

Why is the notation dy/dx not considered a fraction?

Because it represents a limit Signup and view all the answers

What is the term used to describe the process of finding the derivative of a function?

Differentiation Signup and view all the answers

What is the purpose of the formula ∂Gradient at a point = lim(h → 0) [f(a + h) - f(a)]/h?

To determine the gradient of the tangent to a curve at a point Signup and view all the answers

What is the derivative of a function f(x) written as?

f'(x) Signup and view all the answers

What does the notation Dy denote?

The derivative of y with respect to x Signup and view all the answers

What is the term used to describe the process of finding the derivative of a function?

Differentiation from first principles Signup and view all the answers

What is the differential operator denoted by?

All of the above Signup and view all the answers

What does the notation dp/dx mean?

p differentiated with respect to x Signup and view all the answers

What is the importance of being consistent in using notations for derivatives?

To avoid confusion Signup and view all the answers

What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

The derivative is the gradient of the tangent to the graph Signup and view all the answers

What is the primary purpose of using the formula for the derivative to determine the gradient of a tangent to a curve?

To find the expression for the gradient of the graph at any point. Signup and view all the answers

If we have a function ( f(x) = x^2 ), what is the significance of the notation ( f'(x) )?

It represents the rate of change of the function with respect to ( x ). Signup and view all the answers

Which of the following notations is NOT a valid way to denote the derivative of a function ( f(x) )?

( \int f(x) dx ) Signup and view all the answers

What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

The derivative of a function is equal to the gradient of the tangent to the graph. Signup and view all the answers

What is the purpose of the differential operator ( D ) in the notation ( Df(x) )?

To indicate the differentiation of the function. Signup and view all the answers

If we have a function ( f(x) = x^2 ) and we want to find the derivative of the function with respect to ( x ), what is the correct notation to use?

( rac{df}{dx} ) Signup and view all the answers

What is the significance of the notation ( rac{dy}{dx} ) in the context of differentiation?

It represents the derivative of the function with respect to ( x ). Signup and view all the answers

Why is it important to be consistent in using notations for derivatives?

To avoid confusion and ensure clarity in mathematical expressions. Signup and view all the answers

Gr12 Mathematics: Ch 5.2 Differentiation from first Principles

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Questions and Answers

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

What is the purpose of differentiation from first principles?

What is the meaning of the notation dy/dx?

What is the name given to the symbols D and d/dx?

What is the formula for the gradient of a tangent to a curve at a point x = a?

What is the common notation used to denote the derivative of a function f(x)?

Why is the notation dy/dx not considered a fraction?

What is the term used to describe the process of finding the derivative of a function?

What is the purpose of the formula ∂Gradient at a point = lim(h → 0) [f(a + h) - f(a)]/h?

What is the derivative of a function f(x) written as?

What does the notation Dy denote?

What is the term used to describe the process of finding the derivative of a function?

What is the differential operator denoted by?

What does the notation dp/dx mean?

What is the importance of being consistent in using notations for derivatives?

What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

What is the primary purpose of using the formula for the derivative to determine the gradient of a tangent to a curve?

If we have a function ( f(x) = x^2 ), what is the significance of the notation ( f'(x) )?

Which of the following notations is NOT a valid way to denote the derivative of a function ( f(x) )?

What is the relationship between the derivative of a function and the gradient of the tangent to the graph?

What is the purpose of the differential operator ( D ) in the notation ( Df(x) )?

If we have a function ( f(x) = x^2 ) and we want to find the derivative of the function with respect to ( x ), what is the correct notation to use?

What is the significance of the notation ( rac{dy}{dx} ) in the context of differentiation?

Why is it important to be consistent in using notations for derivatives?

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