Master the Derivative of Tan(x): Step-by-Step Guide

d/dx(tanx)

To find the derivative of tan(x) with respect to x, we can use the following steps:

Step 1: Recall the identity for the derivative of tan(x)

To find the derivative of tan(x) with respect to x, we can use the following steps:

Step 1: Recall the identity for the derivative of tan(x).
The derivative of tan(x) is given by:
d/dx(tan(x)) = sec^2(x)

Step 2: Rewrite tan(x) using the identity for tan(x).
tan(x) = sin(x)/cos(x)

Step 3: Differentiate both the numerator and denominator separately using the quotient rule.
To find the derivative of sin(x), we can use the chain rule. The derivative of sin(x) is cos(x).
To find the derivative of cos(x), we can use the chain rule. The derivative of cos(x) is -sin(x).

Differentiating the numerator (sin(x)):
d/dx(sin(x)) = cos(x)

Differentiating the denominator (cos(x)):
d/dx(cos(x)) = -sin(x)

Step 4: Apply the quotient rule to find the derivative of tan(x).
Using the quotient rule, we have:
d/dx(tan(x)) = (cos(x) * cos(x) – (-sin(x) * sin(x))) / cos^2(x)

Simplifying the numerator and the denominator:
= (cos^2(x) + sin^2(x)) / cos^2(x)

Recall the Pythagorean identity:
sin^2(x) + cos^2(x) = 1

Substituting in this identity:
= 1 / cos^2(x)

Finally, using the reciprocal identity:
= sec^2(x)

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

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