A Guide to Finding the Derivative of Tan(x) Using the Quotient Rule

derivative of tanx

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. The quotient rule states that if we have a function in the form of f(x) = g(x)/h(x), the derivative of f(x) is given by:

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

In this case, g(x) = sin(x) and h(x) = cos(x). Therefore, we have:

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

Now let’s find the derivative d(tan(x))/dx using the quotient rule:

[tan(x)]’ = [sin(x) / cos(x)]’

Using the quotient rule, we have:

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

Simplifying this expression, we get:

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

Now, we can simplify further by using the trigonometric identity:

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

[tan(x)]’ = [sin(x) * cos(x) + 1 – cos^2(x)] / (cos(x))^2

Simplifying, we get:

[tan(x)]’ = sin(x) / cos^2(x)

Finally, we can rewrite this in terms of the secant function, which is the reciprocal of the cosine function:

[tan(x)]’ = sin(x) * sec^2(x)

Therefore, the derivative of tan(x) is sin(x) * sec^2(x).

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