A.3 Derivatives of sin⁡𝑥 and cos⁡𝑥

On this screen we're going to develop formally the derivatives of sin⁡𝑥 and cos⁡𝑥, which we will need again and again. We'll use Desmos to help make sense of the results, and of course also practice using them in various Practice Problems below.

In case you'd simply like to memorize these results so you can move on:

Many students find it helpful to remember that, as it turns out, the trig functions that start with "co," like "cosine," have a negative sign in their derivatives, while the trig functions without "co" at their start do not. Hence we can quickly remember that the derivative of cosine is negative sine, while the derivative of sine is (positive) cosine.

You can see a full list of the trig function derivatives on our Handy Table of Trig Function Derivatives, quickly accessible from the "Key Formulas" drop-down menu toward the upper right of every screen. We'll develop the results for sin and cos below, and for the other trig functions later in this Chapter once we have more tools.

I. Derivative of sin⁡𝑥: 𝑑𝑑𝑥sin⁡𝑥 =cos⁡𝑥

To find the derivative of sin⁡𝑥, we start, as always, with the definition of the derivative applied to the function. Recall the definition of the derivative:

We have 𝑓(𝑥) =sin⁡(𝑥). Hence 𝑓(𝑥+ℎ)=sin⁡(𝑥+ℎ)=sin⁡𝑥cos⁡ℎ+sin⁡ℎcos⁡𝑥 where we used the sine addition formula sin⁡(𝑥 +𝑦) =sin⁡𝑥cos⁡𝑦 +sin⁡𝑦cos⁡𝑥. (All such trig formulas are available in our Handy Table of Trig Formulas and Identities, also available from the Key Formulas drop-down menu in the upper right of every screen.)

Making these substitutions in the definition of the derivative for 𝑓(𝑥) =sin⁡𝑥 gives us

You might recognize the "Special Trig Limits" we learned when we were first exploring limits:

The actual reason we introduced those then is because we need them now to complete our proof, continuing from the line marked (*):

That is, we have the key result:

The following Exploration let's you explore this relationship graphically.

II. Derivative of cos⁡𝑥: 𝑑𝑑𝑥cos⁡𝑥 = −sin⁡𝑥

The approach to finding the derivative of cos⁡𝑥 is very much the same as that for sin⁡𝑥 above; we'll just make use of a different trig addition formula.

We start of course with the definition of the derivative applied to 𝑔(𝑥) =cos⁡𝑥. Then 𝑔(𝑥+ℎ)=cos⁡(𝑥+ℎ)=cos⁡𝑥cos⁡ℎ−sin⁡𝑥sin⁡ℎ where we have used the trig addition formula cos⁡(𝑥 +𝑦) =cos⁡𝑥cos⁡𝑦 −sin⁡𝑥sin⁡𝑦.

The definition of the derivative for 𝑔(𝑥) =cos⁡𝑥 then gives us

The following Exploration lets you see the graphical relationship between the function 𝑔(𝑥) =cos⁡𝑥 and its derivative 𝑔′(𝑥) = −sin⁡𝑥.

Practice Problems

As we said above, we'll use these two derivatives again and again and again, so you'll have them memorized soon enough. Let's do a few quick problems now to start:

Practice Problem 4

An equation for the tangent line to the curve 𝑓(𝑥) =cos⁡𝑥 at 𝑥 =𝜋4 is

(A) 𝑦−√22=√22(𝑥−𝜋4)(B) 𝑦−√22=−√22(𝑥−𝜋4) (C) 𝑦+√22=−√22(𝑥−𝜋4)(D) 𝑦−√22=√22(𝑥−𝜋4) (E) none of these

✓✘A✓✘B✓✘C✓✘D✓✘E

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There are two steps to solving this problem:
1. Find the slope of the tangent line 𝑚tangent =𝑓′(𝑥) at the point of interest, 𝑥0 =𝜋4.
2. Find the y-value the at the point of interest 𝑦0 =𝑓(𝜋4). Then write down the equation in Point Slope form 𝑦 −𝑦0 =𝑚tangent(𝑥 −𝑥0).
Step 1: Find the slope of the function 𝑚tangent =𝑓′(𝑥) at the point 𝑥0 =𝜋4.
We found this value in the preceding problem. For completeness, here's that calculation again: 𝑓(𝑥)=cos⁡𝑥𝑓′(𝑥)=−sin⁡𝑥𝑓′(𝜋4)=−sin⁡(𝜋4)=−√22 Step 2: Find the y-value of the point of interest, 𝑦0 =𝑓(𝜋4) =cos⁡(𝜋4): 𝑦0=cos⁡(𝜋4)=√22 Hence we know that the tangent line has slope 𝑚tangent =𝑓′(𝜋/4) = −√22 and passes through the point (𝑥0,𝑦0) =(𝜋4,√22). Its equation in point-slope form is then 𝑦−𝑦0=𝑚tangent(𝑥−𝑥0)𝑦−√22=−√22(𝑥−𝜋4)⟹ (B)✓

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These two derivatives complete our initial toolkit of derivatives: You can now find the derivative of power functions, polynomials, exponential, and the two fundamental trig functions. In the next section we'll see how to take the derivative of a function that is the product or quotient of two other functions, things like 𝑥𝑒𝑥 and sin⁡𝑥cos⁡𝑥.

The Upshot

1. The derivative of sin⁡𝑥 is 𝑑𝑑𝑥sin⁡𝑥 =cos⁡𝑥.
2. The derivative of cos⁡𝑥 is 𝑑𝑑𝑥cos⁡𝑥 = −sin⁡𝑥.

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