Question Video: Finding the First Derivative of a Function Involving a Logarithmic Function Mathematics • Higher Education

Video Transcript

Find the d𝑦 by d𝑥 if 𝑦 is equal to two 𝑥 minus three times the natural logarithm of 𝑥.

We’re given an expression for 𝑦 in terms of 𝑥. And we need to use this to determine d𝑦 by d𝑥; that’s the derivative of 𝑦 with respect to 𝑥. To do this, we need to notice that 𝑦 is the difference between two functions which we know how to differentiate with respect to 𝑥. That means we can evaluate this derivative term by term. First, we recall that d𝑦 by d𝑥 would be the derivative of 𝑦 with respect to 𝑥. That means we need to differentiate two 𝑥 minus three times the natural logarithm of 𝑥 with respect to 𝑥.

To evaluate this derivative, we need to recall one of our rules for differentiation. We can evaluate derivatives term by term. In other words, if 𝑢 of 𝑥 and 𝑣 of 𝑥 are differentiable functions, then the derivative of 𝑢 of 𝑥 minus 𝑣 of 𝑥 with respect to 𝑥 is equal to d𝑢 by d𝑥 minus d𝑣 by d𝑥. And that’s exactly what we have in this case. 𝑢 of 𝑥 is two 𝑥 and 𝑣 of 𝑥 is three times the natural logarithm of 𝑥. So instead of trying to evaluate this derivative directly, we can instead split this into two smaller derivatives, the derivative of two 𝑥 with respect to 𝑥 minus the derivative of three times the natural logarithm of 𝑥 with respect to 𝑥.

And remember, we need to take the difference between these derivatives because we’re taking the difference between their functions. Now, we need to evaluate both of these derivatives. Let’s start with the derivative of two 𝑥 with respect to 𝑥. There’s several different ways we could do this. For example, we could rewrite this as two times 𝑥 to the first power and then use the power rule for differentiation. However, it’s easier to remember that the derivative of any linear function will be the coefficient of 𝑥. In this case, the coefficient of 𝑥 is equal to two. So we can evaluate this derivative to give us two.

To evaluate our second derivative, we need to recall one of our rules for differentiating logarithmic functions. We know for any real constant 𝑎 the derivative of 𝑎 times the natural logarithm of 𝑥 with respect to 𝑥 is equal to 𝑎 divided by 𝑥. In our case, the value of 𝑎 is equal to three. So we set 𝑎 equal to three, meaning we got three divided by 𝑥. So we need to subtract three divided by 𝑥 from two. This gives us two minus three divided by 𝑥, and this gives us our final answer. Therefore, we were able to show if 𝑦 is equal to two 𝑥 minus three times the natural logarithm of 𝑥, then d𝑦 by d𝑥 will be equal to two minus three divided by 𝑥.