If y = (sec x)/(tanx), then dy/dx = ?
To find the derivative of y with respect to x, which is denoted as dy/dx, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2
Now let's apply the quotient rule to find dy/dx for y = (sec x)/(tanx).
To start, we need to differentiate the numerator and denominator separately. The derivative of sec x is obtained using the chain rule, and the derivative of tan x is a simple application of the derivative of trigonometric functions:
Numerator:
dy/dx = (d(sec x)/dx * tan x - sec x * d(tan x)/dx) / (tan x)^2
Next, we differentiate sec x and tan x:
Secant function derivative:
d(sec x)/dx = sec x * tan x
Tangent function derivative:
d(tan x)/dx = sec^2 x
Substituting these derivatives back into the numerator expression, we obtain:
dy/dx = (sec x * tan x * tan x - sec x * sec^2 x) / (tan x)^2
Simplifying further, we have:
dy/dx = (sec x * tan^2 x - sec^3 x) / (tan^2 x)
So, the derivative of y with respect to x, dy/dx, is given by (sec x * tan^2 x - sec^3 x) / (tan^2 x).