# Video: APCALC03AB-P1A-Q13-790101592638

If π is given by the equation π(π₯) = 3π₯Β³ β 3π₯ β ln π₯, what is the slope of the line that is tangent to the graph of π at π₯ = 1.

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### Video Transcript

If π is given by the equation π of π₯ equals three π₯ cubed minus three π₯ minus the natural logarithm of π₯, what is the slope of the line that is tangent to the graph of π at π₯ equals to one.

First, letβ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 π prime of π₯, using differentiation. And then, evaluating that derivative at the given π₯-value.

Letβs find the first derivative of our equation π of π₯ then. We can differentiate the first two terms using the power rule of differentiation. The derivative of three π₯ cubed is three multiplied by three π₯ squared. The derivative of negative three π₯ is negative three. And then, we need to recall how to find the derivative of a natural logarithm. Well, the derivative with respect to π₯ of the natural logarithm of π₯ is actually equal to one over π₯. So, the derivative of negative the natural logarithm of π₯ is negative one over π₯. We just multiply by a constant of negative one.

We, therefore, have an expression for the first derivative π prime of π₯. Itβs equal to nine π₯ squared minus three minus one over π₯. Now, this is the general expression for the slope of the curve at all points for which that slope is defined. But weβve been specifically asked about the slope of the line that is tangent to the graph at a given point. Weβve been asked about the slope when π₯ is equal to one. So, we need to evaluate this first derivative π prime of π₯ when π₯ is equal to one.

Substituting π₯ equals one gives π prime of one is equal to nine multiplied by one squared minus three minus one over one. Thatβs just nine minus three minus one, which is equal to five. So, by applying differentiation, weβve found that the slope of the curve π of π₯ when π₯ is equal to one and, hence, the slope of the line that is tangent to the curve when π₯ is equal to one is five.