Derivative of f(x) = tan(x) | Understanding and Applying the Quotient Rule

d/dx tanx

To find the derivative of the function f(x) = tan(x), we use the quotient rule

To find the derivative of the function f(x) = tan(x), we use the quotient rule. The quotient rule states that if we have a function of 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

In this case, g(x) = sin(x) and h(x) = cos(x). Therefore, we need to find the derivatives g'(x) and h'(x) before we can apply the quotient rule.

The derivative of g(x) = sin(x) is g'(x) = cos(x). This is a basic trigonometric derivative.

Similarly, the derivative of h(x) = cos(x) is h'(x) = -sin(x). Again, this is a basic trigonometric derivative.

Now, we can apply the quotient rule to find the derivative of f(x) = tan(x):

f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2

Expanding the numerator, we get:

f'(x) = (cos^2(x) + sin^2(x)) / [cos(x)]^2

Using the fundamental trigonometric identity sin^2(x) + cos^2(x) = 1, the numerator simplifies to:

f'(x) = 1 / [cos(x)]^2

Recall that tan(x) is defined as the ratio of sin(x) to cos(x). Using this definition, we can rewrite the derivative as:

f'(x) = 1 / [cos(x)]^2 = 1 / (1/[tan(x)])^2 = [tan(x)]^2

Therefore, the derivative of f(x) = tan(x) is f'(x) = [tan(x)]^2.

More Answers: