Use implicit diff. to find dy/dx of each of the following. In the following x,y and (a) are all variables. Show step by step please! Thank you!
1) y^2 = x^2+a^2
2) y^2+ay = x^2+ax+a^2
2 y dy/dx = 2 x + 2 a da/dx
y dy/dx = (x+a da/dx)
dy/dx = (x + a da/dx)/y
or
dy/dx = (x + a da/dx)/(x^2+a^2)^.5
---------------------------------
2 y dy/dx + a dy/dx + y da/dx = 2 x^2 + a + x da/dx + 2 a da/dx
(2y+a)dy/dx = 2 x^2 +a +(x-y+2a)da/dx
dy/dx = [2 x^2 +a +(x-y+2a)da/dx]/(2y+a)
I suspect in each, a is a constant, so da/dx=0
I figured the same but he said specifically that a was variable.
My answers were like these
1)
2yy'=2x+2a
y'=2x+2a/2y
2)
2yy'+ay'+y=2x+a+x+2a-y
2yy'+ay'=3x+3a-y
y'(2y+a)=3x+3a-y
y'=3x+3a-y/2y+a
And my teacher said I made mistakes with my answers but I couldn't figure out.
To find the derivative dy/dx using implicit differentiation, you'll follow these steps:
Step 1: Differentiate each term of the equation with respect to x.
Step 2: Treat y as a function of x and use the chain rule to differentiate terms containing y.
Step 3: Solve the resulting equation for dy/dx.
Let's apply these steps to each of the given equations:
1) y^2 = x^2 + a^2
Step 1: Differentiate each term:
2y * (dy/dx) = 2x
Step 2: Treat y as a function of x:
2 * y * dy/dx = 2x
Step 3: Solve for dy/dx:
dy/dx = 2x / [2 * y]
dy/dx = x / y
Therefore, the derivative dy/dx for the first equation is dy/dx = x / y.
2) y^2 + ay = x^2 + ax + a^2
Step 1: Differentiate each term:
2y * (dy/dx) + a * (dy/dx) = 2x + 2a
Step 2: Treat y as a function of x:
2 * y * dy/dx + a * dy/dx = 2x + 2a
Step 3: Solve for dy/dx:
dy/dx * (2y + a) = 2x + 2a
dy/dx = (2x + 2a) / (2y + a)
Therefore, the derivative dy/dx for the second equation is dy/dx = (2x + 2a) / (2y + a).
By following the steps of implicit differentiation, we can find the derivatives dy/dx for each equation.