# Find \frac{d^{69}}{dx^{69}} \cos(-1.5x).

## Question:

Find {eq}\displaystyle \frac{d^{69}}{dx^{69}} \cos(-1.5x) {/eq}.

## Derivatives of Sine and Cosine:

When taking the derivative of either of the two fundamental trigonometry functions, {eq}sin(x) {/eq} and {eq}cos(x) {/eq}, it makes it much easier to find the solution when you know the pattern that arises when you keep taking the derivative. Here is the pattern:

{eq}\frac{d}{dx}(\sin(x)) = \cos(x) \\ \frac{d}{dx}(\cos(x)) = -\sin(x) \\ \frac{d}{dx}(-\sin(x)) = -\cos(x) \\ \frac{d}{dx}(-\cos(x)) = \sin(x) {/eq}.

Notice that it repeats after every 4 derivatives. This is important because if were are asked to take the derivative many times, we can just divide the number of times by 4. In this case, it's not the quotient that would matter, but the remainder would. The remainder tells us how many more we need to take to arrive at our answer.

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We are being asked to find {eq}\displaystyle \frac{d^{69}}{dx^{69}} \cos(-1.5x) {/eq}.

Because taking the derivative repeats after every 4...

Calculating Derivatives of Trigonometric Functions

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Chapter 8 / Lesson 3
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Trigonometric functions appear almost everywhere that there is a repeating pattern. Discover how to calculate the derivatives of trigonometric functions and learn how to graph them.