If y = 2x^3 - 4x and dx/dt = 4, find dy/dt when x = 1.
Please show me how to solve this problem with steps.
first find dy/dx
dy/dx = 6x^2 - 4
then dy /dx for x = 1
dy/dx = 6(1)^2 - 4
dy/dx = 2
then apply chain rule
dy/dt = dy/dx * dx/dt
dy/dt = 2 * 4
dy/dt = 8
just differentiate with respect to t
dy/dt = 6x^2 dx/dt - 4 dx/dt
sub in dx/dt=4 and x=1
dy/dt = 6(1)(4) - 4(4)
= 24 - 16 = 8
To find dy/dt when x = 1, we need to differentiate y with respect to t using the chain rule.
Step 1: Find dy/dx by differentiating y with respect to x.
Given, y = 2x^3 - 4x
To find dy/dx, differentiate each term with respect to x:
d/dx(2x^3) = 6x^2 (using the power rule for differentiation)
d/dx(-4x) = -4 (using the constant multiple rule)
So, dy/dx = 6x^2 - 4.
Step 2: Use the chain rule to find dy/dt.
Given, dx/dt = 4.
We want to find dy/dt when x = 1.
Using the chain rule:
dy/dt = (dy/dx) * (dx/dt)
Substituting the known values:
dy/dt = (6x^2 - 4) * (dx/dt)
= (6(1)^2 - 4) * 4
= (6 - 4) * 4
= 2 * 4
= 8.
So, dy/dt = 8 when x = 1.
To solve this problem, we will use the chain rule, which states that for a composition of functions u(x) = f(g(x)), the derivative of u(x) with respect to x is given by du/dx = f'(g(x)) * g'(x).
Given the function y = 2x^3 - 4x, we want to find dy/dt when x = 1.
1. First, differentiate the function y = 2x^3 - 4x with respect to x to find dy/dx.
dy/dx = d/dx (2x^3 - 4x)
= 6x^2 - 4
2. Secondly, since we are given dx/dt = 4, we need to find dx/dt when x = 1.
dx/dt = 4 (given)
3. Now, we can use the chain rule to find dy/dt. Multiply the derivatives dy/dx and dx/dt:
dy/dt = (dy/dx) * (dx/dt)
= (6x^2 - 4) * (dx/dt)
4. Finally, substitute x = 1 and dx/dt = 4 into the equation we obtained in step 3:
dy/dt = (6(1)^2 - 4) * 4
= (6 - 4) * 4
= 2 * 4
= 8
Therefore, dy/dt = 8 when x = 1.