How to Find the Derivative of tan(x) with Respect to x Using the Quotient Rule

d/dx tanx

To find the derivative of the function f(x) = tan(x) with respect to x (denoted as d/dx tan(x)), we can use the quotient rule

To find the derivative of the function f(x) = tan(x) with respect to x (denoted as d/dx 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), where g(x) and h(x) are both differentiable functions, then the derivative is given by:

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

Applying this rule to our function f(x) = tan(x), we need to identify g(x) and h(x). In this case, g(x) is simply 1 because tan(x) = sin(x)/cos(x), and h(x) is cos(x).

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

g'(x) = d/dx (1) = 0 (since the derivative of any constant is zero)

h'(x) = d/dx (cos(x)) = -sin(x) (using the derivative of cos(x) which is -sin(x))

Plugging these into the quotient rule, we have:

f'(x) = (0 * cos(x) – 1 * (-sin(x))) / [cos(x)]^2
= sin(x) / cos^2(x)

Simplifying further, we know that sin(x) / cos^2(x) can be rewritten as sin(x) * (1/cos^2(x)), which is equal to sin(x) * sec^2(x), where sec(x) is the reciprocal of cos(x).

Therefore, the derivative of tan(x) with respect to x is:

d/dx tan(x) = f'(x) = sin(x) * sec^2(x)

Note: It’s worth mentioning that there is an alternative approach to finding the derivative of tan(x) using trigonometric identities. By writing tan(x) as sin(x)/cos(x), we can rewrite it as (1/cos(x)) * sin(x), which is sec(x) * sin(x). The derivative of this expression can be found using the product rule and will yield the same result as above, sin(x) * sec^2(x).

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