Exploring the Derivative of the Secant Function Using Quotient and Chain Rules in Calculus

derivative of secant

The derivative of the secant function can be found by using the quotient rule and the chain rule in calculus

The derivative of the secant function can be found by using the quotient rule and the chain rule in calculus.

The secant function is defined as sec(x) = 1/cos(x), where cos(x) is the cosine function.

To find the derivative of sec(x), we can rewrite it as sec(x) = (1/cos(x)) * (cos(x)/cos(x)) = (cos(x))/(cos²(x)).

Now, let’s differentiate this expression using the quotient rule.

The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), then its derivative is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x))/(h(x))^2.

Applying the quotient rule to our expression for sec(x), we have:

(f(x))’ = ((cos(x))’ * (cos²(x)) – (cos(x)) * (cos²(x))’)/((cos²(x))^2).

The derivative of the cosine function is -sin(x), and the derivative of the square of the cosine function is -2cos(x)sin(x). Substituting these derivatives into our expression, we get:

(f(x))’ = (-sin(x) * (cos²(x)) – (cos(x)) * (-2cos(x)sin(x)))/((cos²(x))^2).

Simplifying further, we have:

(f(x))’ = (-sin(x)cos²(x) + 2cos²(x)sin²(x))/((cos²(x))^2).

Now, we can simplify the numerator by factoring out a sin(x) term:

(f(x))’ = (sin(x) * (2cos²(x)sin(x) – cos²(x)))/((cos²(x))^2).

Finally, we can simplify the expression by canceling out the common factor of sin(x):

(f(x))’ = (2cos²(x)sin(x) – cos²(x))/(cos^4(x)).

And there you have it! The derivative of the secant function is (2cos²(x)sin(x) – cos²(x))/(cos^4(x)).

More Answers: