Question

use implicit differentiation to find the slope of the tangent line to the curve

3^x + log_2(xy) = 10

at the point (2,1) and use it to find the equation of the tangent line in the form y = mx + b

Answer 1

im not really sure how to differentiate this if anyone could explain that to me

Answer 2

your differeantiation is

3^3*ln(3) + 1/(xy)*{1+xdy/dx} = 0

find the value of dy/dx form here, put the value of m = dy/dx on slope, and put the value of x,y on the that eqn of tangent, find the value of b, you have your tangent

Answer 3

i really don't think i know how to solve for dy/dx my professor just started implicit differentiation and I'm lost on this stuff

Answer 4

okay differentiate this equation

x^2+y=0

Answer 5

2x+1=0

Answer 6

no ... it's 2x+dy/dx = 0,

here y is a variable that depends on x,

Answer 7

of course you can see that, y = -x^2, which is a parabola,
on this particular case, you can separate y, but this is not always the case

Answer 8

ok so still how exactly do i get dy/dx by itself

Answer 9

differentiate this wrt x

y = -x^2

Answer 10

you can differentiate y using chain rule,

for eg you have x^2+y^2 = 25, try differentiating this

Answer 11

2x+2y(dy/dx=0

Answer 12

2x+2y(dy/dx)=0

Answer 13

yes, so you have differential equation,
though you don't know that is y, you can find slope of tangent at given point

okay, what is slope of tangent at (0,5)

Answer 14

the actual act of differentiating its figuring out how to separate the dy/dx to the other side in order to solve for it

Answer 15

i wouldn't say solving it though, you can find the function for slope of tangent of given curve

Answer 16

I'm sorry, I'm not sure what you mean

Answer 17

f(x,y) = 0 gives a curve in x,y plane .. i guess you know that

Answer 18

so I'm getting the slope as -10.05 when i plug in my x and y

Answer 19

yeah, you get it because you know the FUNCTION OF SLOPE, and
PUTTING x,y in FUNCTION GIVES the value of function, which is slope


and also, you know that dy/dx gives the tangent ... right,

so it must the the slope of tangent ... now you know it!!!