Implicit Differentiation HELP PLEASE. I keep getting the wrong answer?

Question

Use implicit differentiation to find the slope tangent line to the curve (x^3y^4)+(10y^5)= (x^5) + 335 at the point (1,2).

The Slope M = ?

I keep getting -75/816, and it's wrong. Can anyone help?

Answer 1

Use the product rule when differentiating the 1st term and you get:

x^3*4y^3*dy/dx + 3x^2*y^4 + 50y^4*dy/dx = 5x^4

Plug in 1 for x and 2 for y:

32*dy/dx + 48 + 800*dy/dx = 5

832*dy/dx = -43

dy/dx = -43/832 = m

Answer 2

What you're doing is basically differentiating everything with respect to x, then solving for (dy/dx).

A common mistake many students make is forgetting that the x-derivative of y is (dy/dx), and not 1. For example, when you're differentiating something like y^3, it becomes 3y^2 * (dy/dx) by the chain rule, NOT just 3y^2.

With that, let's take a look:

We use the product rule on the leftmost term, the chain rule on both the terms on the left, and differentiate the terms on the right as normal:

x^3*4y^3*(dy/dx) + 3x^2*y^4 + 50y^4*(dy/dx) = 5x^4;
(dy/dx)*[4x^3*y^3 + 50y^4] = 5x^4 - 3x^2*y^4;

(dy/dx) = [5x^4 - 3x^2*y^4] / [4x^3*y^3 + 50y^4].

Plugging in x = 1 and y = 2 gives us

(dy/dx) (1,2) = [5 - 3*16] / [4*8 + 50*16] = -43/832.


Note that solving for (dy/dx) is just like solving for any other variable: get all the terms that have it on one side of the equation, factor it out, and divide away any terms OTHER than (dy/dx) to isolate it.

Answer 3

3x^2y^4 + x^3(3y^3) y' + 50y^4 y' = 5x^4
Letting x = 1 and y = 2 and solving for y'
I get y ' = -43/824

Answer 4

to reply to this question "implicitly" you do here: Re-Write the function: (9x+y)^a million/2 = 2 + [(x^2)(y^2)] Differential Implicitly: (a million/2(9x+y)^-a million/2)(9+a million[dy/dx]) = (x^2)(2y[dy/dx]) + (y^2)(2x) circulate all [dy/dx] words to the left area of the equation: [dy/dx] = (y^2)(2x)(9/2[9x+y]^-a million/2) / (x^2)(2y)