find the differential dy of the given function y=3x-4 sec(2x-1)
To find the differential dy of the given function y = 3x - 4 sec(2x - 1), we will take the derivative with respect to x.
Step 1: Differentiate the first term, 3x, with respect to x:
d/dx(3x) = 3
Step 2: Differentiate the second term, -4 sec(2x - 1), with respect to x using the chain rule:
d/dx[-4 sec(2x - 1)] = -4 * sec(2x - 1) * tan(2x - 1) * d/dx(2x - 1)
Step 3: Differentiate (2x - 1) with respect to x:
d/dx(2x - 1) = 2
Step 4: Substitute the above derivative back into step 2:
d/dx[-4 sec(2x - 1)] = -4 * sec(2x - 1) * tan(2x - 1) * 2
Step 5: Simplify the expression:
d/dx[-4 sec(2x - 1)] = -8 sec(2x - 1) tan(2x - 1)
Step 6: Combine the derivatives from both terms:
dy/dx = 3 - 8 sec(2x - 1) tan(2x - 1)
So, the differential dy of the given function y = 3x - 4 sec(2x - 1) is dy/dx = 3 - 8 sec(2x - 1) tan(2x - 1).