Find dy/dx by implicit differentiation.
y^5 + x^2y^3 = 1 + x^4y
So, first I find the derivative:
5y^4 + 2x3y^2 = 4x^3(1)
Now I find dy/dx:
But this is where I don't know what to do?
Please Help.
Did I do the derivative part wrong?
Should it be:
5y^4 + (x^2 . 3y^2 + y^3 . 2x) = (x^4 . 1 + y . 4x^3)
you appear to be very very confused.
y^5 + (x^2)(y^3) = 1 + (x^4)(y)
remember you are finding dy/dx, so you are differentiating with respect to x
Which means that when you differentiate a y term you will have dy/dx tagging along
5y^4 dy/dx + (x^2)(3y^2)dy/dx + (y^3)(2x) = 0 + x^4 dy/dx + y(4x^3)
get all dy/dx terms to the left side
5y^4 dy/dx + 3x^2(y^2) dy/dx - x^4 dy/dx = 4y(x^3) - 2x(y^3)
dy/dx = (4y(x^3) - 2x(y^3)) / (5y^4 + 3x^2(y^2) - x^4 )
check my typing, easy to make an error with all those exponents
To find dy/dx, we can differentiate both sides of the equation with respect to x using the chain rule when necessary.
Differentiating the left side of the equation, we have:
d/dx (y^5 + x^2y^3) = d/dx (1 + x^4y)
=> d/dx (y^5) + d/dx (x^2y^3) = d/dx (1) + d/dx (x^4y)
Using the power rule, we can differentiate each term separately.
For the first term, d/dx (y^5), we apply the chain rule:
d/dx (y^5) = 5y^4 (dy/dx)
For the second term, d/dx (x^2y^3):
Using the product rule, we get:
d/dx (x^2y^3) = 2x(1)(y^3) + x^2(3y^2(dy/dx))
= 2xy^3 + 3x^2y^2 (dy/dx)
For the right-hand side of the equation, d/dx (1) = 0 since the derivative of a constant is zero.
For d/dx (x^4y), we also use the product rule:
d/dx (x^4y) = x^4 (dy/dx) + 4x^3y
Now we can substitute these derivatives back into the original equation:
5y^4(dy/dx) + 2xy^3 + 3x^2y^2 (dy/dx) = x^4 (dy/dx) + 4x^3y
Next, let's combine the terms with dy/dx on one side to solve for dy/dx:
5y^4(dy/dx) + 3x^2y^2 (dy/dx) - x^4 (dy/dx) = -2xy^3 - 4x^3y
Factoring out the common factor of (dy/dx), we have:
(5y^4 + 3x^2y^2 - x^4)(dy/dx) = -2xy^3 - 4x^3y
Finally, we can solve for dy/dx:
dy/dx = (-2xy^3 - 4x^3y) / (5y^4 + 3x^2y^2 - x^4)
Therefore, the derivative dy/dx is given by (-2xy^3 - 4x^3y) / (5y^4 + 3x^2y^2 - x^4).
To find dy/dx using implicit differentiation, start by differentiating both sides of the equation with respect to x.
Let's go step by step:
1. Differentiate y^5 with respect to x:
To differentiate y^5, use the chain rule. Multiply by the derivative of the exponent, which is dy/dx.
The derivative of y^5 with respect to x is 5y^4(dy/dx).
2. Differentiate x^2y^3 with respect to x:
To differentiate x^2y^3, use the product rule. Differentiate each term separately.
The derivative of x^2 is 2x.
The derivative of y^3 with respect to x is 3y^2(dy/dx).
3. Differentiate 1 with respect to x:
The derivative of any constant, like 1, is zero.
4. Differentiate x^4y with respect to x:
To differentiate x^4y, use the product rule. Differentiate each term separately.
The derivative of x^4 is 4x^3.
The derivative of y with respect to x is dy/dx.
Combining all the derivatives, the equation becomes:
5y^4(dy/dx) + 2x(y^3)(dy/dx) = 4x^3 + 4x^3(dy/dx)
Now, let's bring all the dy/dx terms to one side of the equation and the other terms on the other side:
5y^4(dy/dx) + 2x(y^3)(dy/dx) - 4x^3(dy/dx) = 4x^3
To simplify, factor out dy/dx:
dy/dx (5y^4 + 2xy^3 - 4x^3) = 4x^3
Finally, divide both sides by (5y^4 + 2xy^3 - 4x^3) to solve for dy/dx:
dy/dx = 4x^3 / (5y^4 + 2xy^3 - 4x^3)
And that's the derivative dy/dx using implicit differentiation for the given equation.