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Differentiate implicitly and then determine the equation of the tangent line to the curve defined by the given equation at the indicated point.$$x^{2}+y^{3}=1+\ln \left(x^{2}+1\right)+\ln y(0,1)$$

$$y=1$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:30

Find an equation of the ta…

02:32

Find an equation of the st…

02:00

01:41

01:11

Find the equation of the t…

01:13

01:00

Find the slope of the tang…

right. So we're asked to find the equation of the Tanja line. If you're actually do the tangent line, you need a point and a slope. Um, now the point they give you in the problem in the directions is 01 Uh, and then the slope that they give you they don't give you that's selling for D y DX at the ordered pair. 01 Um, yeah, that's all we need. So we already have half the information. What we need to do now is find the derivative implicitly of the problem to give you x squared. Plus y cubed equals one plus the natural log of X word plus one plus natural log of why here we go. So if we do the, uh, implicit differentiation derivative of exporters to x the derivative of y cube, just three y squared the yds, the derivative of one is zero. So you don't need to write that down. The derivative of natural law. Will That function goes into the denominator and in the numerator goes derivative um, plus again one over. Why do y dx So as I can what I would do. This is just a me thing is I wouldn't solve the do i. D. X. I think that's too much work. I would start plugging in zeros for exes. This is kind of nice, because too damn 00 and zero divide by one. There's a zero right here is well, eso we have several zeros in the problem. And when I plug one and for why one squared, it's one times three So I have three d y d X equals again zero plus one d y dx um so from here just I don't know subtract the y dx over so d do to do I d. X equals zero and you divide by two and you figure out that slope At that point, I'll switch back to black is equal to zero is the only way to get two times something to equal zeros. If there's something in zero eso now, we have enough information. Give us the equation. Are you know our point or archangel line? Sorry, I'll write this in green. I like to do point slope form So why it goes a slope and then x minus your x coordinate plus your y coordinate. The only thing is zero times. Anything is nothing. So you can just simplify that to be Why equals one? It's a horizontal change. A line right there.

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