Mastering the Derivative of Tan(x): Simplifying with the Quotient Rule and Sec^2(x)

Derivative of tan(x)

sec^2u du

The derivative of tan(x) is sec^2(x).

To find the derivative of tan(x), we need to use the quotient rule since tan(x) is a quotient of sin(x) and cos(x):

f(x) = sin(x)
g(x) = cos(x)

tan(x) = f(x)/g(x)

Using the quotient rule formula:

[f'(x)g(x) – f(x)g'(x)]/[g(x)]^2

f'(x) = cos(x)
g'(x) = -sin(x)

Substituting the values of f'(x), g'(x), f(x), and g(x) into the formula:

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

Simplifying:

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

Recall that:

sin^2(x) + cos^2(x) = 1

Thus:

tan'(x) = 1/cos^2(x)

And because:

sec^2(x) = 1/cos^2(x)

Then:

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

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