Let y = tan(5x+2).
Find the differential dy when x = 2 and dx = 0.4
Find the differential dy when x = 2 and dx = 0.8
To find the differential dy, we can use the derivative of y with respect to x and multiply it by the given differential dx.
First, let's find the derivative of y = tan(5x+2) with respect to x. We can use the chain rule to differentiate the function.
Step 1: Rewrite the function using the identity tan(u) = sin(u) / cos(u)
y = sin(5x+2) / cos(5x+2)
Step 2: Apply the chain rule to differentiate each term separately.
dy/dx = [(cos(5x+2))(5) - (sin(5x+2))(5)] / cos^2(5x+2)
dy/dx = 5(cos(5x+2) - sin(5x+2)) / cos^2(5x+2)
Now, let's substitute x = 2 into the derivative equation and find dy when dx = 0.4 and dx = 0.8.
When x = 2 and dx = 0.4:
dy = 5(cos(5(2)+2) - sin(5(2)+2)) / cos^2(5(2)+2) * dx
dy = 5(cos(12) - sin(12)) / cos^2(12) * 0.4
When x = 2 and dx = 0.8:
dy = 5(cos(5(2)+2) - sin(5(2)+2)) / cos^2(5(2)+2) * dx
dy = 5(cos(12) - sin(12)) / cos^2(12) * 0.8
Now, calculate the values of dy using a calculator or software with the given values of x and dx.