If 𝑓 is given by the equation 𝑓 of 𝑥 equals three 𝑥 cubed minus three 𝑥 minus the natural logarithm of 𝑥, what is the slope of the line that is tangent to the graph of 𝑓 at 𝑥 equals to one.
First, let’s recall what it means for a line to be tangent to a graph at a point. The tangent to a curve at a given point has the same slope as the curve itself at that point. We can find the slope of a curve at any given point and, hence, the slope of the tangent to the curve at any given point by finding its first derivative, in this case 𝑓 prime of 𝑥, using differentiation. And then, evaluating that derivative at the given 𝑥-value.
Let’s find the first derivative of our equation 𝑓 of 𝑥 then. We can differentiate the first two terms using the power rule of differentiation. The derivative of three 𝑥 cubed is three multiplied by three 𝑥 squared. The derivative of negative three 𝑥 is negative three. And then, we need to recall how to find the derivative of a natural logarithm. Well, the derivative with respect to 𝑥 of the natural logarithm of 𝑥 is actually equal to one over 𝑥. So, the derivative of negative the natural logarithm of 𝑥 is negative one over 𝑥. We just multiply by a constant of negative one.
We, therefore, have an expression for the first derivative 𝑓 prime of 𝑥. It’s equal to nine 𝑥 squared minus three minus one over 𝑥. Now, this is the general expression for the slope of the curve at all points for which that slope is defined. But we’ve been specifically asked about the slope of the line that is tangent to the graph at a given point. We’ve been asked about the slope when 𝑥 is equal to one. So, we need to evaluate this first derivative 𝑓 prime of 𝑥 when 𝑥 is equal to one.
Substituting 𝑥 equals one gives 𝑓 prime of one is equal to nine multiplied by one squared minus three minus one over one. That’s just nine minus three minus one, which is equal to five. So, by applying differentiation, we’ve found that the slope of the curve 𝑓 of 𝑥 when 𝑥 is equal to one and, hence, the slope of the line that is tangent to the curve when 𝑥 is equal to one is five.