Mastering the Quotient Rule: A Step-by-Step Guide to Finding the Derivative of tan(x)

𝑑/𝑑𝑥[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 sin(x)/cos(x):

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

Now, let’s apply the quotient rule:

d/dx [sin(x)/cos(x)] = (cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]) / (cos(x))^2

The derivative of sin(x) with respect to x is cos(x), and the derivative of cos(x) with respect to x is -sin(x):

d/dx [sin(x)] = cos(x)
d/dx [cos(x)] = -sin(x)

Plugging these values into the quotient rule formula:

d/dx [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)

Since cos^2(x) is equal to 1/(sec^2(x)), we can simplify the expression:

d/dx [tan(x)] = 1 / cos^2(x) = 1 / (1/sec^2(x)) = sec^2(x)

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

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