The video tutorial explains the concept of the gradient function and its notation, including f dash and dy/dx. It introduces derivatives, their origin, and how they relate to the gradient function. The tutorial demonstrates using first principles to calculate the gradient and simplify expressions to find the derivative. It clarifies the differentiation process and its terminology, emphasizing the difference between deriving and differentiating. Finally, the video applies the derivative to solve problems, such as finding the equation of a tangent line.

What notation should be used when a question is provided in function notation?

• d y on dx

• g of x

• f dash of x

• y prime

What is another name for the gradient function?

• Limit

• Derivative

• Integral

• Slope

What is the first step in using first principles to find the gradient function?

• Substitute x with x plus h

• Evaluate the limit

• Expand the brackets

• Write down the first full line of first principles

What should you be cautious of when expanding brackets in first principles?

• Getting the signs wrong

• Forgetting to multiply by h

• Using the wrong function

• Incorrectly applying the limit

What is the correct term for the process of finding the gradient function?

• Calculating

• Differentiating

• Deriving

• Integrating

Why should you not use the term 'deriving' when finding the gradient function?

• It is not used in calculus

• It is not a recognized term

• It is a different mathematical process

• It is too similar to 'integrating'

What is the gradient at x equals two for the function discussed?

• 4

• -4

• 0

• 2

What are the three components needed to find the equation of a line?

• Two points and a slope

• An x-coordinate, a y-coordinate, and a gradient

• A tangent, a secant, and a point

• A function, a derivative, and a limit

What is the significance of x equals two in the parabola discussed?

• It is the vertex

• It is the axis of symmetry

• It is the y-intercept

• It is the maximum point

What does the equation of the tangent at point P represent?

• A diagonal line

• A horizontal line

• A vertical line

• A curved line