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

Find the derivative of 7π₯ + 4 sin π₯ with respect to cos π₯ + 1 at π₯ = π/6.

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### Video Transcript

Find the derivative of seven π₯ plus four sin π₯ with respect to cos π₯ plus one at π₯ equals π by six.

Letβs begin by defining our two functions. Weβll let π¦ be equal to seven π₯ plus four sin π₯. And weβll define cos π₯ plus one as π§. We then recall that, given two parametric equations β π₯ equals π of π‘ and π¦ equals π of π‘ β we can find dπ¦ by dπ₯ by multiplying dπ¦ by dπ‘ by one over dπ₯ by dπ‘. Or equivalently, by dividing dπ¦ by dπ‘ by dπ₯ by dπ‘.

Now in this case, our two functions are π¦ and π§. And they are in terms of π₯. So we can see that dπ¦ by dπ§ must be equal to dπ¦ by dπ₯ divided by dπ§ by dπ₯. So weβre going to need to begin by differentiating each of our functions with respect to π₯. The first derivative of seven π₯ is seven. And when we differentiate sin π₯, we get cos π₯. So we see that dπ¦ by dπ₯ here is seven plus four cos of π₯.

We also know that if we differentiate cos of π₯, we get negative sin of π₯. So thatβs dπ§ by dπ₯. Itβs negative sin π₯. dπ¦ by dπ§ is the quotient. Itβs seven plus four cos of π₯ divided by negative sin π₯. But weβre not quite finished. Weβre looking to find the derivative at the point where π₯ is equal to π by six. So weβre going to substitute π₯ for π by six in our expression. Thatβ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, thatβs the same as multiplying the numerator by two. And so we see that the derivative of our function, seven π₯ plus four sin π₯, with respect to cos of π₯ plus one at π₯ equals π by six is negative 14 minus four root three.