implicit differentiation (1 Viewer)

[pLuvia]

Given x^3 + 3x^2y^4 + x4y^2 - 6y = 0 then differentiate both sides with respect to x

can someone help me with this?

thx

 

[FinalFantasy]

Given x^3 + 3x^2y^4 + x4y^2 - 6y = 0 then differentiate both sides with respect to x

can someone help me with this?

thx

3x²+(6x)(y^4)+4y³y'(3x²)+4y²+8yy'x-6y'=0
then get y' as the subject
y'=dy\dx

 

[pLuvia]

could you show me please, first time learning it.. sorry

 

[Trev]

Why are you attempting this question when you haven't learnt implicit differentiation yet? Wait till you do extension 2.

 

[FinalFantasy]

Implicit Differentiation:
Just the same as "normal" differentiation except whenever u differentiate the y, u add a y' next to it as well
then u get y' as the subject at the end

 

[Trev]

x² + y² = 1
Implicity with respect to x gives:
2x + 2y(dy/dx) = 0
dy/dx = -(x/y)
There's a basic example for you...

 

[AFGHAN22]

Is implicit differentiation 4 unit maths or 3 unit maths? Cause we r doin it in year 11!!!!!!!!!!

 

[AFGHAN22]

Thanx fr the example trev can u please solve this
x squared plus 2xy minis y squared

 

[Trev]

4 unit.
10char

 

[pLuvia]

x^2 + 2xy - y^2

d/dx x^2 = 2x
d/dx 2xy = 2 ydy/dx
d/dx -y^2 = -2y dydx

2x + 2 y dy/dx -2y dy/dx

= 2x

is that right?

dy/dx = 2x

 

[Trev]

x²+2xy-y² (=0?)
2x + x(dy/dx) + y - 2y(dy/dx) = 0
(x-2y)(dy/dx) = -(2x+y)
dy/dx= -(2x+y)/(x-2y)

 

[Slidey]

Why are you attempting this question when you haven't learnt implicit differentiation yet? Wait till you do extension 2.

Damnable lies.

Learn by example, kadlil:

You know that for y=x^3,
y'=3x^2, right?

And you know by chain rule that for y=(x^2-3)^3,
y'=3*(x^2-3)'*(x^2-3)^2=6x(x^2-3)^2, right?

And similarly, in general, for y=(f(x))^3,
y'=3*f'(x)*(f(x))^2, right?

Well, implicit differentiation is actually just using chain rule:
for y=y^3-x, let y=f(x), y=f(x)=(f(x))^3-x
y'=3*f'(x)*(f(x))^2 - 1
y'=3y'*y^2 - 1 (since y=f(x))
Isolate y':
y'-3y'*y^2=-1
y'(1-3y^2)=-1
y'=-1/(1-3y^2)

Here's some quick examples:

x^3+y^3=6xy
dy/dx=y'. We find it by taking the derivative of both sides:
3x^2+3y'y^2=6xy'+6y (chain rule on LHS and product rule on LHS)
3x^2-6y=6xy'-3y'y^2 (put multiples of y' on one side)
3(x^2-2y)=3y'(2x-y^2)
y'=(x^2-2y)/(2x-y^2)

x^2+y^2=r^2 (your generic circle)
2x+2y'y=0
y'y=-x
y'=-x/y

The hyperbola y=1/x can be expressed as 1=xy
1=xy
0=y+xy'
-y/x=y'
BUT, y=1/x, so
y'=-(1/x)/x
y'=-1/x^2
Now let us derive y normally:
y=1/x=x^-1
y'=-x^-2=-1/x^2

 

[Slidey]

Here, try these and tell me your answers, kadlil:

1) sin(y)+cos(x)=y^2

2) y^2e^y=x

3) xy=ln(xy)

4) lny=yx^2

 

[Trev]

Damnable lies.

Learn by example, kadlil:

Haha, yes yes Slide, I'm sorry

 

[Slidey]

Haha. I just think it's good if people can learn as much MX2 as they can in year 11.

 

[dawso]

yeah, topics like this that are technically 4unit, shud be looked at by 3unit as well if they want 2 succeed as they can be used,....just like partial fractions, and some forms of integration, polynomials...um.....thats about it

 

[pLuvia]

Here, try these and tell me your answers, kadlil:

1) sin(y)+cos(x)=y^2

2) y^2e^y=x

3) xy=ln(xy)

4) lny=yx^2

haha sorry slide, i havent learnt the sin, cos, tan stuff, ive only learnt normal ones, up to max, min, tangent, normal etc, physical world stuff. And im learning integration in the physical world, pretty damn easy

im starting to understand thanks Slide Rule u Rule lol

 

[pLuvia]

yeah, topics like this that are technically 4unit, shud be looked at by 3unit as well if they want 2 succeed as they can be used,....just like partial fractions, and some forms of integration, polynomials...um.....thats about it

is implicit differentiation really 4u stuff?? or is it harder 3u questions?

 

[Trev]

is implicit differentiation really 4u stuff?? or is it harder 3u questions?

4u, like I said before... It hasn't changed since I said

 

[pLuvia]

wow, 4u ay, are they simple stuff..? oh by the way when do you use implicit differentiation trev?

simple stuff for 4u that is