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Problem 53 Hard Difficulty

Find an equation of the tangent line to the curve at the given point.
$ y = \sin (\sin x), (\pi, 0) $

Answer

$y=-x+\pi$

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Video Transcript

Let's find the equation of the tangent line to the curve y equals sign of sine X at the point. Pi zero. The slope of the tangent line will be the derivative at that point. So let's find the derivative. And we need to use the chain rule because we have an inside function and an outside function. It's a composite function, so the derivative of the outside the derivative of Sign is co sign. So we've co sign of sine X times. The derivative of the inside The inside is sign so times coastline X. Now we want the derivative evaluated at pi. We don't have to simplify. First, we can go ahead and substitute the pie in there right now. And we have co sign of signed pie Times CO sign pie. Well, the sign of pious zero sweep co sign of zero and the coastline of pie is negative. One. The coastline of zero is one, so we one times negative one. So the slope is negative One. Now, To find the equation of the line, we can use point slope form. Why minus y one equals m times X minus X one where the point X one y one is pi zero. So we have y minus zero equals Air Slope negative one times a quantity, X minus pi. Let's simplify that. So we have. Why equals the opposite of X plus high. That's the equation of the tangent line.