How to Find the Derivative of Tan(x) | Step-by-Step Guide and Formula

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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 derivative of tan(x) can be found as follows:

Step 1: Identify the quotient rule formula:
If we have a function f(x) which can be expressed as the quotient of two functions u(x) and v(x), i.e., f(x) = u(x)/v(x), then the derivative of f(x) with respect to x, denoted as f'(x), is given by:
f'(x) = (u'(x)*v(x) – u(x)*v'(x)) / [v(x)]^2

Step 2: Apply the quotient rule to tan(x):
Let’s express tan(x) as the quotient of sin(x) and cos(x), i.e., tan(x) = sin(x)/cos(x).
In this case, u(x) = sin(x) and v(x) = cos(x).

Step 3: Find the derivatives of u(x) and v(x):
To find the derivative of sin(x) with respect to x, we can use the chain rule, which states that the derivative of sin(x) is cos(x).

u'(x) = cos(x)

Similarly, to find the derivative of cos(x) with respect to x, we use the chain rule, which states that the derivative of cos(x) is -sin(x).

v'(x) = -sin(x)

Step 4: Apply the quotient rule formula:
Using the quotient rule formula, we can now find the derivative of tan(x):

tan'(x) = (u'(x)*v(x) – u(x)*v'(x)) / [v(x)]^2
= (cos(x)*cos(x) – sin(x)*(-sin(x))) / [cos(x)]^2
= (cos^2(x) + sin^2(x)) / [cos(x)]^2 (using the trigonometric identity sin^2(x) + cos^2(x) = 1)
= 1/[cos(x)]^2

Step 5: Simplify the expression:
The expression 1/[cos(x)]^2 can be simplified further using the reciprocal identity of cosine (sec(x)), which states that 1/cos(x) = sec(x).

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

So, the derivative of tan(x) with respect to x is represented as tan'(x) = sec^2(x).

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