The Quotient Rule | Finding the Derivative of tan(x) with Step-by-Step Explanation

d/dx [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 for two functions u(x) and v(x), if their derivatives are u'(x) and v'(x) respectively, then the derivative of their quotient u(x)/v(x) is given by:

(u(x)v'(x) – v(x)u'(x)) / v(x)^2

In this case, u(x) is tan(x) and v(x) is 1.

So, we have:

u(x) = tan(x)

v(x) = 1

Taking the derivatives:

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

v'(x) = 0 (since the derivative of a constant is always zero)

Now plugging the values into the quotient rule:

(tan(x) * 0 – 1 * sec^2(x)) / 1^2

Simplifying further:

-sec^2(x) / 1

Finally, we can simplify the expression to:

-dx/sec^2(x)

So, the derivative of tan(x) is -sec^2(x) or alternatively, -dx/sec^2(x).

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