If y = 5x^3 − 10x, determine [dy/dt] when x = 0 and [dx/dt] = −2 .
help please ;)
dy/dt=15x^2*dx/dt-10dx/dt
To find [dy/dt], we need to use the chain rule, as we are dealing with the derivative of a function of two variables, y = f(x). Here's the step-by-step process:
1. Differentiate the function y = 5x^3 - 10x with respect to x to get dy/dx. This will give us the derivative of y with respect to x.
dy/dx = d/dx (5x^3 - 10x)
Using the power rule for differentiation, the derivative of x^n is n*x^(n-1), where n is a constant:
dy/dx = 15x^2 - 10
2. Given dx/dt = -2, we need to find dt/dx to use the chain rule properly. Rearranging the equation gives us dt = dx / (-2). Therefore, dt/dx = -1/(2dx).
3. Finally, we can apply the chain rule to find [dy/dt].
[dy/dt] = (dy/dx) * (dt/dx)
Substituting the values we determined earlier:
[dy/dt] = (15x^2 - 10) * (-1/(2dx))
Since we are given that x = 0 and dx/dt = -2:
[dy/dt] = (15(0)^2 - 10) * (-1/(2(-2)))
Simplifying further, we obtain:
[dy/dt] = (-10) * (-1/(-4))
[dy/dt] = 10/4
Therefore, when x = 0 and dx/dt = -2, [dy/dt] is equal to 10/4, or 2.5.