Derivative of tan(x) using the Quotient Rule

derivative of tan(x)

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.

Let’s start by writing tan(x) as the ratio of sine(x) and cosine(x):

tan(x) = sin(x) / cos(x)

Now, we can use the quotient rule for differentiation, which states that if we have a function f(x) = g(x) / h(x), then the derivative f'(x) is given by:

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

Applying this rule to tan(x), we have:

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

Let’s find the derivatives of sin(x) and cos(x):

sin'(x) = cos(x)

cos'(x) = -sin(x)

Now, substitute these values back into the equation:

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)]

Recall the trigonometric identity: sec^2(x) = 1 / cos^2(x)

Therefore, we can express the derivative of tan(x) as:

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

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

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