Question Video: Differentiating a Combination of Trigonometric and Linear Functions at a Point Mathematics • Higher Education

Video Transcript

Find the derivative of seven 𝑥 plus four sin 𝑥 with respect to cos 𝑥 plus one at 𝑥 equals 𝜋 by six.

Let’s begin by defining our two functions. We’ll let 𝑦 be equal to seven 𝑥 plus four sin 𝑥. And we’ll define cos 𝑥 plus one as 𝑧. We then recall that, given two parametric equations — 𝑥 equals 𝑓 of 𝑡 and 𝑦 equals 𝑔 of 𝑡 — we can find d𝑦 by d𝑥 by multiplying d𝑦 by d𝑡 by one over d𝑥 by d𝑡. Or equivalently, by dividing d𝑦 by d𝑡 by d𝑥 by d𝑡.

Now in this case, our two functions are 𝑦 and 𝑧. And they are in terms of 𝑥. So we can see that d𝑦 by d𝑧 must be equal to d𝑦 by d𝑥 divided by d𝑧 by d𝑥. So we’re going to need to begin by differentiating each of our functions with respect to 𝑥. The first derivative of seven 𝑥 is seven. And when we differentiate sin 𝑥, we get cos 𝑥. So we see that d𝑦 by d𝑥 here is seven plus four cos of 𝑥.

We also know that if we differentiate cos of 𝑥, we get negative sin of 𝑥. So that’s d𝑧 by d𝑥. It’s negative sin 𝑥. d𝑦 by d𝑧 is the quotient. It’s seven plus four cos of 𝑥 divided by negative sin 𝑥. But we’re not quite finished. We’re looking to find the derivative at the point where 𝑥 is equal to 𝜋 by six. So we’re going to substitute 𝑥 for 𝜋 by six in our expression. That’s seven plus four cos of 𝜋 by six over negative sin of 𝜋 by six.

Cos of 𝜋 by six is root three over two. And sin of 𝜋 by six is one-half. When we divide by one-half, that’s the same as multiplying the numerator by two. And so we see that the derivative of our function, seven 𝑥 plus four sin 𝑥, with respect to cos of 𝑥 plus one at 𝑥 equals 𝜋 by six is negative 14 minus four root three.