Find the Derivative - d/dx y=(sec(x))/x

Problem

d()/d(x)sec(x)/x

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two terms, sec(x) and x use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=sec(x) and v=x

3. Differentiate the individual components. The derivative of the numerator is d(sec(x))/d(x)=sec(x)*tan(x) and the derivative of the denominator is d(x)/d(x)=1

4. Substitute these values into the quotient rule formula.

d(y)/d(x)=(x*sec(x)*tan(x)−sec(x)⋅1)/(x2)

5. Factor the numerator to simplify the expression. Extract the common term sec(x)

d(y)/d(x)=(sec(x)*(x*tan(x)−1))/(x2)

Final Answer

d()/d(x)sec(x)/x=(sec(x)*(x*tan(x)−1))/(x2)

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