   Chapter 3.5, Problem 28E

Chapter
Section
Textbook Problem

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.28. x2 + 2xy + 4y2 = 12, (2, 1) (ellipse)

To determine

To find: The equation of the tangent line to the given equation at the point.

Explanation

Given:

The curve is x2+2xy+4y2=12.

The point is (2,1).

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then

dydx=dydududx.

(2) Product rule: If y=f(u) and u=g(x)  are both differentiable function, then

ddx(f(x)+g(x))=ddx(f(x))+ddx(g(x)).

Formula used:

The equation of the tangent line at (x1,y1) is, yy1=m(xx1) (1)

where, m is the slope of the tangent line at (x1,y1) and m=dydx|x=x1,y=y1.

Calculation:

Consider the equation x2+2xy+4y2=12.

Differentiate the above equation implicitly with respect to x,

ddx(x2+2xy+4y2)=ddx(12)ddx(x2)+ddx(2xy)+ddx(4y2)=0ddx(x2)+2ddx(xy)+4ddx(y2)=0

Apply the product rule (2),

2x+2[xddx(y)+yddx(x)]+4ddx(y2)=02x+2[xdydx+y(1)]+4ddx(y2)=0

Apply the chain rule (1) and simplify the terms,

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