Implicit differentiation is a way to differentiate an expression by treating one of the variables as a function of the other. You should be using chain rule to answer these problems. I'll walk you through the necessary steps.
1. Differentiate both sides of equation (d/dx):
d/dx (3x^2-2y^2) = d/dx (5)
6x (dx/dx) - 4y (dy/dx) = 0 *Note that you don't have to write dx/dx in your step (I just included it to help your understanding*
2. Rearrange the equation to solve for dy/dx:
4y (dy/dx) = 6x
dy/dx = 6x/4y = 3x/2y
3. Differentiate both sides of equation again (d/dx):
d/dx (dy/dx) = d/dx (3x/2y) <-- Use a quotient rule
d^2y/dx^2 = ((3*2y) - (3x*2*dy/dx))/(4y^2)
4. Replace dy/dx with the expression solved earlier:
d^2y/dx^2 = ((3*2y) - (3x*2*3x/2y))/(4y^2)
= 3*2y*y - 3x*3x/(4y^3)
= (6y^2-9x^2)/(4y^3)
= -3(3x^2-2y^2)/(4y^3) <--- The question told us that 3x^2-2y^2 = 5
= -3*5/(4y^3)
= -15/(4y^3)
I hope this answers your question. Please let me know if you have any further questions.
Hi, thank you for your question.
Implicit differentiation is a way to differentiate an expression by treating one of the variables as a function of the other. You should be using chain rule to answer these problems. I'll walk you through the necessary steps.
1. Differentiate both sides of equation (d/dx):
d/dx (3x^2-2y^2) = d/dx (5)
6x (dx/dx) - 4y (dy/dx) = 0 *Note that you don't have to write dx/dx in your step (I just included it to help your understanding*
2. Rearrange the equation to solve for dy/dx:
4y (dy/dx) = 6x
dy/dx = 6x/4y = 3x/2y
3. Differentiate both sides of equation again (d/dx):
d/dx (dy/dx) = d/dx (3x/2y) <-- Use a quotient rule
d^2y/dx^2 = ((3*2y) - (3x*2*dy/dx))/(4y^2)
4. Replace dy/dx with the expression solved earlier:
d^2y/dx^2 = ((3*2y) - (3x*2*3x/2y))/(4y^2)
= 3*2y*y - 3x*3x/(4y^3)
= (6y^2-9x^2)/(4y^3)
= -3(3x^2-2y^2)/(4y^3) <--- The question told us that 3x^2-2y^2 = 5
= -3*5/(4y^3)
= -15/(4y^3)
I hope this answers your question. Please let me know if you have any further questions.