Chapter 3: The Derivative

Question 1

Week 3

Deduce the equation of the derivative.

Answer 1

We know from the definition of the derivative that it is the slope of the tangent to the curve at a certain point.

We also know that the average rate of change is the slope of the secant line between two points P and Q in an interval.

Well, if we want to compute the slope of the tangent at point P, we can make Q approach P. In other words, we can make the interval between P and Q (known as h) reach 0

We already know the formula for the average rate of change.

Δy/Δx = f(x+h) - f(x) / h

so, if we add a limit function, we transform the equation to lim (h –> 0) f(x) = f(x+h) - f(x) / h

This equation, in essence, is the mathematical definition of the derivative.

Question 2

Week 5

What is the definition of a derivative

Answer 2

Formula

lim_{h -> 0} f(x+h) - f(x)/h

Alternative Formula

lim_{z -> x} f(x) - f(x)/z-x

z = x + h
h = z - x

When is a Function Differentiable?

The derivative of a function is also the slope of the tangent line to a point on f(x)

A function is only differentiable if.

lim_{h -> 0+} f(x+h) - f(x)/h

is equal to

lim_{h -> 0-} f(x+h) - f(x)/h

Question 3

Week 5

State common differentiation rules.

Answer 3

• Constant Rule
• Positive Integer Rule
• Constant Multiple Rule
• Sum Rule
• Product Rule
• Quotient Rule

Question 4

Week 5

What are higher order derivatives?

Answer 4

• f’(x) = first derivative
• f’‘(x) = second derivative
• f’’‘(x) = third derivative