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Find the slope of the tangent line

Find dy/dx for x2y+ √2x + 5y=7
Show that the point (x,y) = (2,1) lies on the curve defined by the equation above and find the slope of the tangent line at this point.

What part is under the square-root?
2x+5y=7 is under the square root

Michael J. | Effective High School STEM Tutor & CUNY Math Peer LeaderEffective High School STEM Tutor & CUNY ...
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Use implicit differentiation using the chain rule to find dy/dx.  Your derivative will be in terms of x and y.  Plug in x=2 and x=1 to evaluate dy/dx.  This will give you the slope of the tangent line.

Mark M. | Math Tutor--High School/College levelsMath Tutor--High School/College levels
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x2y4 + (2x + 5y)½ = 7

Plugging in x = 2 and y = 1, we get 7 = 7.  So, the point (2,1) lies on the curve.

Implicitly differentiate the given function to obtain:

2xy4 + 4y3(dy/dx)(x2) + (½)(2x + 5y)(2 +5dy/dx) = 0

Plug in x = 2 and y = 1:

4 + 16(dy/dx) + (1/6)(2 + 5dy/dx) = 0

4 + 16(dy/dx) + 1/3 + (5/6)(dy/dx) = 0

(101/6)(dy/dx) = -13/3

dy/dx = (-13/3)(6/101) = -26/101