The Derivative of Tan(x) with Respect to x | Step-by-Step Explanation and Trigonometric Identity

(d/dx) tanx

To find the derivative of tan(x) with respect to x, we can use the chain rule

To find the derivative of tan(x) with respect to x, we can use the chain rule. The derivative of the tangent function is given by:

(d/dx) tan(x) = sec^2(x)

Here’s the step-by-step explanation of how we get this result:

1. Start with the definition of the tangent function: tan(x) = sin(x)/cos(x).

2. Differentiate both the numerator and denominator separately. We use the quotient rule for differentiation.

– The derivative of sin(x) with respect to x is cos(x).
– The derivative of cos(x) with respect to x is -sin(x).

Applying the quotient rule, we get:

(d/dx) [sin(x)/cos(x)] = (cos(x) * cos(x) – sin(x) * (-sin(x))) / cos^2(x)

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

Since sin^2(x) + cos^2(x) = 1, the expression becomes:
= 1 / cos^2(x)

3. Using the trigonometric identity sec^2(x) = 1 / cos^2(x), we have:

(d/dx) tan(x) = sec^2(x)

And that’s the derivative of tan(x) with respect to x.

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