Learn How to Find the Derivative of tan(x) with Respect to x | Step-by-Step Guide with Examples

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

To find the derivative of tan(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), where both g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:

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

In the case of tan(x), we can write it as tan(x) = sin(x) / cos(x). Applying the quotient rule, we have:

tan'(x) = ((cos(x) * cos(x)) – (sin(x) * (-sin(x)))) / (cos(x))^2
= (cos^2(x) + sin^2(x)) / (cos^2(x))
= 1 / (cos^2(x))

Therefore, the derivative of tan(x) with respect to x is 1 / (cos^2(x)).

Another approach to finding the derivative of tan(x) is to use the identity tan(x) = sin(x) / cos(x), and express tan(x) in terms of sin(x) and cos(x). Then, we can use the chain rule to differentiate. Here’s how it would look:

tan(x) = sin(x) / cos(x)
tan(x) = sin(x) * cos^(-1)(x)

Now, we can consider tan(x) as the product of sin(x) and cos^(-1)(x), and apply the product rule:

tan'(x) = sin(x) * (-1) * (cos^(-2)(x)) * sin(x) + cos^(-1)(x) * cos(x)
= -sin^2(x) * cos^(-2)(x) + cos^(-1)(x) * cos(x)

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we have:

tan'(x) = -1 * cos^(-2)(x) + cos^(-1)(x) * cos(x)
= -1 * cos^(-2)(x) + cos^(-1)(x) / cos(x)
= -1 * cos^(-2)(x) + 1 / (cos(x) * cos(x))
= -1 * cos^(-2)(x) + 1 / cos^2(x)
= -1 * cos^(-2)(x) + 1 / cos^2(x)
= 1 / cos^2(x) – 1 / cos^2(x)
= 1 / cos^2(x)

Therefore, the derivative of tan(x) with respect to x is 1 / cos^2(x).

In summary, both approaches yield the same result: the derivative of tan(x) with respect to x is 1 / cos^2(x).

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