Let = tan(5x + 3) Find the differential dy when x = 5 and dx = .3 Find the differential dy when x = 5 and dx =.6

Question

Let = tan(5x + 3)

Find the differential dy when x = 5 and dx = .3

Find the differential dy when x = 5 and dx =.6

Answer 1

You've undoubtedly been instructed that:

So, that means (multiplying both sides of the equation by "dx"):

dy = dx[f '(x)]

Using the rules for taking derivatives of trig functions, if f(x) = tan(5x + 3), then f '(x) = 5sec²(5x + 3)

That means (since dx = 0.3 and x = 5) dy = 0.3[5sec²(5(5) + 3)] = 1.5sec²(28). Since your calculator doesn't have a secant button, you must rely on the fact that sec(θ) = 1/cos(θ) so:

1.5sec²(28) = 1.5/cos²(28). Make sure you are in "radian mode" and you get cos(28) = -0.9626 so cos²(28) = 0.9266 and 1.5/0.9266 = 1.619

For x = 5 and dx = 0.6, just plug those value in for "x" and "dx" into dy = dx[5sec²(5x + 3)]