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

If 𝑦 = 2𝑥 + sin 3𝑥, find d𝑦/d𝑥.

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Video Transcript

If 𝑦 is equal to two 𝑥 plus the sin of three 𝑥, find the derivative of 𝑦 with respect to 𝑥.

We need to find an expression for the derivative of 𝑦 with respect to 𝑥. And we can see in this instance 𝑦 is given as the sum of two functions. It’s two 𝑥 plus the sin of three 𝑥. And in actual fact, we know how to differentiate both of these two terms individually. This means we can just evaluate this derivative term by term. This means we can split d𝑦 by d𝑥, which is the derivative of two 𝑥 plus the sin of three 𝑥 with respect to 𝑥, into the derivative of two 𝑥 with respect to 𝑥 plus the derivative of the sin of three 𝑥 with respect to 𝑥.

There’s several different ways of evaluating the derivative of two 𝑥 with respect to 𝑥. For example, we could write this as two 𝑥 to the first power and use the power rule for differentiation. However, we also know that two 𝑥 is a linear function. And we know the slope of a linear function is just the coefficient of 𝑥. In other words, for any real constant 𝑎, the derivative of 𝑎𝑥 with respect to 𝑥 is just equal to 𝑎. So instead of using the power rule for differentiation, we can just take the coefficient of 𝑥, which in this case is two.

To evaluate the derivative of our second term, we need to notice this is a standard trigonometric derivative result which we should commit to memory. We know for any real constant 𝑎, the derivative of the sin of 𝑎𝑥 with respect to 𝑥 is equal to 𝑎 times the cos of 𝑎𝑥. In this case, our coefficient of 𝑥 is equal to three. So we’ll set 𝑎 equal to three.

And so by applying this rule with our value of 𝑎 equal to three, we get three times the cos of three 𝑥. And this is our final answer. Therefore, we were able to show if 𝑦 is equal to two 𝑥 plus the sin of three 𝑥, then d𝑦 by d𝑥 will be equal to two plus three times the cos of three 𝑥.