How do you find dy/dx by implicit differentiation with respect to x?
x^3 + y^5 = 1
To find dy/dx by implicit differentiation, you can follow these steps:
1. Differentiate both sides of the equation with respect to x, treating y as a function of x:
d/dx (x^3 + y^5) = d/dx (1)
2. For the left side, you need to use the chain rule. Differentiate each term with respect to x:
d/dx (x^3) + d/dx (y^5) = 0
3. For the first term, you only need to apply the power rule for differentiation:
3x^2 + d/dx (y^5) = 0
4. For the second term, you need to apply the chain rule. Differentiate y^5 with respect to y and then multiply by dy/dx:
3x^2 + 5y^4 (dy/dx) = 0
5. Now, you can isolate dy/dx by moving all the terms involving dy/dx to one side and solve for it:
5y^4 (dy/dx) = -3x^2
dy/dx = -3x^2 / (5y^4)
So, the derivative dy/dx is -3x^2 / (5y^4).