Find an equation of the tangent line to the graph of the function at the given point. x e^y+y e^

step 1 All right, so equation of a tangent line. What you need is a point and a slope. So if you get in

Find an equation of the tangent line to the graph of the...

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Key Concepts

Tangent Line Equation

The tangent line equation represents the line that touches a curve at a given point and has the same slope as the curve at that point. It is typically found using the point-slope form of a line, which utilizes the coordinates of the point of tangency along with the derivative (slope) of the function at that point.

Implicit Differentiation

Implicit differentiation is a method used to find the derivative of a function when it is not explicitly solved for one variable. This technique involves differentiating both sides of an equation with respect to one variable while treating the other variable as an implicit function of that variable, and then solving for the derivative.

Product Rule

The product rule is a fundamental differentiation rule applied when two functions are multiplied together. It states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Chain Rule

The chain rule is used for differentiating composite functions. It allows one to take the derivative of a function that is composed of another function by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

Transcript

00:02 All right, so equation of a tangent line.

00:06 What you need is a point and a slope.

00:10 So if you get in the equation, it makes perfect sense then that you can use point slope form to find a tangent line.

00:18 So they give you the point already is 0 .1.

00:23 And let's see, i would write it this way.

00:27 To find the slope, you need, d, y, dx, when at the ordered pair, 01.

00:34 And once you find that, then you can get the answer.

00:38 So they give you the equation x, e to the y.

00:43 So this is going to be implicit differentiation, the way it's set of y, e to the x is equal to 1.

00:51 And each one of these is the product rule, where the derivative of x is one, leave e to the y alone, plus leave x alone.

01:00 The derivative of e to the y is e to the y times d, y, y, d, y, y, x.

01:04 And then plus the derivative of y is d y d x e to the x power plus leave y alone the derivative of e to the x is e to the x is equal to zero so what's nice about this is anywhere you have x is zero like right here that'll equal zero so i'm going to ignore that piece as i'm looking at this you will you have to be careful because e to the zero power is equal to one so that's just d -y -d -x times one.

01:40 And this is another, you know, one times one.

01:44 So that's plus one there.

01:45 And then e to the first power is equal to e...

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