Chapter 2
Computing Derivatives
2.1 Elementary derivative rules
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are alternate notations for the derivative?
• How can we sometimes use the algebraic structure of a function f (x) to easily
compute a formula for f ′ (x)?
• What is the derivative of a power function of the form f (x) = x n ? What is the
derivative of an exponential function of form f (x) = a x ?
• If we know the derivative of y = f (x), how is the derivative of y = k f (x) computed,
where k is a constant?
• If we know the derivatives of y = f (x) and y = g(x), how is the derivative of
y = f (x) + g(x) computed?
Introduction
In Chapter 1, we developed the concept of the derivative of a function. We now know
that the derivative f ′ of a function f measures the instantaneous rate of change of f with
respect to x as well as the slope of the tangent line to y = f (x) at any given value of x.
To date, we have focused primarily on interpreting the derivative graphically or, in the
context of functions in a physical setting, as a meaningful rate of change. To actually
calculate the value of the derivative at a specific point, we have typically relied on the limit
87
Computing Derivatives
2.1 Elementary derivative rules
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are alternate notations for the derivative?
• How can we sometimes use the algebraic structure of a function f (x) to easily
compute a formula for f ′ (x)?
• What is the derivative of a power function of the form f (x) = x n ? What is the
derivative of an exponential function of form f (x) = a x ?
• If we know the derivative of y = f (x), how is the derivative of y = k f (x) computed,
where k is a constant?
• If we know the derivatives of y = f (x) and y = g(x), how is the derivative of
y = f (x) + g(x) computed?
Introduction
In Chapter 1, we developed the concept of the derivative of a function. We now know
that the derivative f ′ of a function f measures the instantaneous rate of change of f with
respect to x as well as the slope of the tangent line to y = f (x) at any given value of x.
To date, we have focused primarily on interpreting the derivative graphically or, in the
context of functions in a physical setting, as a meaningful rate of change. To actually
calculate the value of the derivative at a specific point, we have typically relied on the limit
87