How to Find the Derivative of Tan(x) using the Quotient Rule and Simplifying

d/dx tanx

To find the derivative of tan(x), we can use the quotient rule

To find the derivative of tan(x), 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 of f(x) is given by:

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

Let’s find the derivative of tan(x) using the quotient rule.

First, we need to define the functions g(x) and h(x). In this case, g(x) = sin(x) and h(x) = cos(x).

Now, let’s find the derivatives of g(x) and h(x).

g'(x) = d/dx (sin(x)) = cos(x)

h'(x) = d/dx (cos(x)) = -sin(x)

Now we can substitute these derivatives into the quotient rule formula to find the derivative of tan(x):

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

Simplifying further:

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

We know that sin^2(x) + cos^2(x) = 1, so we can simplify this expression by substituting:

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

Since sec^2(x) = 1 / (cos^2(x)), we can further simplify the expression as:

f'(x) = sec^2(x)

Therefore, the derivative of tan(x) is sec^2(x).

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