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.