How to differentiate the function tan(x) using the quotient rule in calculus

d/dx(tanx)

To differentiate the function tan(x), we can use the quotient rule or rewrite tan(x) in terms of sine and cosine and then differentiate

To differentiate the function tan(x), we can use the quotient rule or rewrite tan(x) in terms of sine and cosine and then differentiate.

Let’s use the quotient rule for differentiation.
The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) with respect to x is given by:

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

In our case, g(x) = sin(x) and h(x) = cos(x).
Therefore, g'(x) = cos(x) (differentiation of sin(x)) and h'(x) = -sin(x) (differentiation of cos(x)).

Applying the quotient rule:
d/dx(tan(x)) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / cos^2(x)
= (cos^2(x) + sin^2(x)) / cos^2(x)
= 1 / cos^2(x)

Finally, we can simplify the expression by using a trigonometric identity:
1 / cos^2(x) = sec^2(x)

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

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