This is not a function but it would still be nice to be able to find the slope Note use of chain rule This cant be solved for y This technique is called implicit differentiation 1 Differentiate both sides wrt ID: 647515
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Slide1
Implicit DifferentiationSlide2
This is not a function, but it would still be nice to be able to find the slope.
Note use of chain rule.Slide3
This can’t be solved for
y
.
This technique is called implicit differentiation.
1 Differentiate both sides w.r.t.
x
.
2 Solve for .Slide4
We need the slope. Since we can’t solve for
y
, we use implicit differentiation to solve for .
Find the equations of the lines tangent and normal to the curve at .
Note product rule.Slide5
Find the equations of the lines tangent and normal to the curve at .
tangent:
normal:Slide6
Examples:
x
2y3 – xy = 10
x2+3xy+y2 = -1
x2+y2 = 25
y3-x2 = x + y3y +
ln y = 4exx
2y= y3x + 5y + x25x2 + 8x – 16y
2 - 4y – 9 = 0Slide7
More examples:
Find dy
/dx
y = sin x + cos y
x2y + y3 + 3 = xy
xy3 + xy = 6
ex + cos y = ln
y6x2 + 2xy + y4
= 1x2y2 – ex + 2y = 4sin x/y = ½
cos (x+y) + sin (x+y) = 1/3
ecos x +
esin y = 1/4