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Find an equation of the tangent line to the curve at the given point.

$ y = x^2 \ln x, (1, 0) $

$y=x-1$

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Idaho State University

in this problem, we have to find the equation for the tangent line to a curve and specifically were given a curve that is defined by a log rhythmic function. So we have the function. Why equals X square times the natural log of X and what we need to do before we confined the tangent line is find my prime. Well, how do we do that? We're going to have to apply the product rule. So why Prime would be equal to d X square dx times the national log of X plus x squared times d natural log of X over d x So when you do that, we simplify just a little bit. We get why prime equals two x times the natural log of X plus x So we have the derivative. But we want to know the equation for the tangent line. So what do we have to do now? We need the slope of our tangent line at the point that we're told in the problem 10 so we can plug in our X value into y prime. So why prime would be equal to two times one times the natural log of one plus one, so we would get zero plus one, which equals one. And now we confined the tangent line very easily. We have this form. Why? Minus why not equals M times X minus X not. And then we could just plug in the X and Y coordinates from our point, so we would have y minus zero equals one times X minus one, and then we can simplify to get the tangent line equals Why equal toe X negative one x minus one. Excuse me. So I hope that this problem helped you understand how we can find the tangent line to a curb using differentiation, specifically finding the derivative of a of a logarithmic function.