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Find an equation of the tangent line to the curve at thegiven point.$y=\sin x+\sin ^{2} x, \quad(0,0)$

$y=x$

Calculus 1 / AB

Chapter 3

Derivatives

Section 5

The Chain Rule

Harvey Mudd College

University of Nottingham

Idaho State University

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we have the function, Why equals sine X plus sign Square X and we want to find the equation of the tangent line to that function at the 0.0 Now the first thing we should check is that 00 is actually a point on the graph of dysfunction. By plugging X equals zero, I get sign of zero, which is zero plus sine squared of zero, which is zero times 00 So I get zero and the 00.0 is indeed on the graph of this of this function. So that's good. So we want the equation of the tent of some tangent line to find the equation of a line in general, it is sufficient to know the slope of the line and a point on the line so that we can then used this standard formula for the equation of a line. Why, minus y one equals and the times x minus X one where X one y one is a point on our line and em is the slope. Well, we already have a point on our line. The tangent line to this function at 00 passes through 00 So we have a point on our line. The only thing we need to find is M the slope of our tangent line and the slope of our tangent line at 00 will be the derivative of this function at X equals zero. So what we should do next is take the derivative of dysfunction. So the derivative of Sine X is co sign X. And now we use the chain rule. We have an inter function sign of X and an outer function and input raised to the power of two by the chain rule. We start with taking the derivative of the outermost layer, which is an input squared. Take the derivative of that by the power we bring the two down, keep the input the same and subtract one from the original exponents that becomes exponents of one and then multiply by the derivative of the input. The inter function sign of X. The derivative of that is co sign next and this is a one in fact, not a prime. So here's our why Prime, We can now plug in zero to get the slope of the tangent line at that point co sign of zero is one plus two times sign of zero is zero and then times one. So this term is zero. So this whole thing equals once our slope is one. So now we just plug in the relevant data into our equation, and it simplifies to y equals X, and that's it.

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