Find the derivative of (6x^2+4y^2)/(3x+7y)=3 using implicit and quotients rule. Thank you!
(6x^2+4y^2)/(3x+7y)=3
we know that (u/v)' = (u'v-uv')/v^2, so
((12x+8yy')(3x+7y)-(6x^2+4y^2)(3+7y'))/(3x+7y)^2 = 0
36x^2 + 84xy + 24xyy' + 56y^2y'-18x^2-12y^2-42x^2y'-28y^2y' = 0
because the denominator is not zero
y'(24x+56y^2-42x^2-28y^2) = 18x^2+12y^2-36x^2-84xy
Now just divide and simplify. check on wolframalpha.com by entering
derivative (6x^2+4y^2)/(3x+7y)=3
Thank you. I appreciate the help.
To find the derivative of the given equation using implicit differentiation, we differentiate both sides of the equation with respect to x and treat y as a function of x.
Step 1: Differentiate both sides of the equation using the product rule and chain rule:
For the left side of the equation: (6x^2+4y^2)
- The derivative of 6x^2 with respect to x is 12x.
- The derivative of 4y^2 with respect to x is 8y * (dy/dx) using the chain rule.
For the right side of the equation: (3x+7y) = 3
- The derivative of 3x with respect to x is 3.
- The derivative of 7y with respect to x is 7 * (dy/dx) using the chain rule.
Step 2: Set the derivatives equal to each other since both sides of the equation are equal.
12x + 8y * (dy/dx) = 3 + 7 * (dy/dx)
Step 3: Rearrange the equation to solve for (dy/dx), the derivative of y with respect to x.
8y * (dy/dx) - 7 * (dy/dx) = 3 - 12x
(8y - 7) * (dy/dx) = 3 - 12x
(dy/dx) = (3 - 12x) / (8y - 7)
Therefore, the derivative of (6x^2+4y^2)/(3x+7y) = 3 using implicit differentiation is (3 - 12x) / (8y - 7).
If you also want to find the derivative using the quotient rule, let's walk through the steps:
The quotient rule states that for a function u(x)/v(x), the derivative can be found as:
(u'(x)v(x) - u(x)v'(x)) / (v(x))^2
In this case, u(x) = 6x^2 + 4y^2 and v(x) = 3x + 7y.
Step 1: Find the derivatives of u(x) and v(x) individually.
The derivative of u(x) = 12x.
To find the derivative of v(x), we treat y as a function of x and use the chain rule.
v'(x) = 3 - 7(dy/dx)
Step 2: Apply the quotient rule using the derivatives of u(x) and v(x):
[(12x)(3x + 7y) - (6x^2 + 4y^2)(3 - 7(dy/dx))] / (3x + 7y)^2
Simplifying the numerator gives: (36x^2 + 84xy - 18x^2 - 42y^2 + 28y(dy/dx))
Step 3: Simplify further if possible.
(18x^2 + 84xy - 42y^2 + 28y(dy/dx)) / (3x + 7y)^2
Finally, we can divide both the numerator and denominator by 14 to simplify the expression:
(9x^2 + 42xy - 21y^2 + 14y(dy/dx)) / (3x + 7y)^2
So, the derivative of (6x^2+4y^2)/(3x+7y)=3 using the quotient rule is (9x^2 + 42xy - 21y^2 + 14y(dy/dx)) / (3x + 7y)^2.