Question Video: Identifying the Graph of a Derivative Mathematics

Video Transcript

Which of the following graphs represents the derivative of 𝑓 of 𝑥 equals 𝑥 cubed plus 𝑥? Option (A), (B), (C), (D), or (E)?

In this question, we need to identify which of five given graphs represents the derivative of the given polynomial function 𝑓 of 𝑥. The easiest way to answer this question is to note that since 𝑓 of 𝑥 is a polynomial, we can differentiate it with respect to 𝑥 term by term by using the power rule for differentiation. We multiply by the exponent of 𝑥 and reduce the exponent by one. We get that 𝑓 prime of 𝑥 equals three 𝑥 squared plus one.

We see that 𝑓 prime of 𝑥 is a quadratic with positive leading coefficient. Therefore, its graph is a parabola that opens upwards. We can see that the constant term is one, so the 𝑦-intercept of the graph must be at one. We can see that this only matches the graph in option (C).

This is enough to answer the question. However, we often want to analyze the derivative of functions we cannot easily differentiate. In cases like this, it is a good idea to consider the properties of 𝑓 of 𝑥 and see how these properties will affect the graph of its derivative. So although we can answer this question by differentiation, let’s try to do this without differentiating the function 𝑓 of 𝑥.

First, we can see that 𝑓 of 𝑥 is the sum of two terms, both of which are increasing functions. We know that the sum of increasing functions is itself increasing. So 𝑓 of 𝑥 is an increasing function. Since the function is increasing, we know that its derivative must be nonnegative. Therefore, the graph of the derivative must not have any section below the 𝑥-axis. This is enough to say that the answer must be option (C).

We can analyze the derivative further by factoring 𝑓 of 𝑥. We see that 𝑓 of 𝑥 equals 𝑥 times 𝑥 squared plus one. We know that 𝑥 squared plus one is positive for any value of 𝑥. So 𝑓 only has a single 𝑥-intercept at the origin. This allows us to sketch the graph of 𝑓 of 𝑥. It is a positive leading coefficient cubic with only one 𝑥-intercept at the origin.

Noting that 𝑓 of 𝑥 is increasing and sketching its graph gives us the following. We see that its slope is always nonnegative, so the only possible answer is option (C).