How to Find the Derivative of the Tangent Function: Step-by-Step Guide with Quotient Rule & Simplification

ddx tanx

To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule. The derivative of tan(x) can be expressed as:

ddx tanx = ddx (sinx/cosx)

Let’s denote u = sinx and v = cosx. Applying the quotient rule, we have:

ddx (sinx/cosx) = (v du/dx – u dv/dx) / (v^2)

Now, let’s find the derivatives of u and v:

du/dx = d/dx (sinx) = cosx
dv/dx = d/dx (cosx) = -sinx

Substituting these values into the quotient rule formula, we get:

ddx (sinx/cosx) = (cosx * cosx – sinx * (-sinx)) / (cosx)^2

Simplifying further:

ddx (sinx/cosx) = (cos^2(x) + sin^2(x)) / (cos^2(x))
= 1 / cos^2(x)

Since tan(x) is defined as sinx/cosx, we can rewrite this result as:

ddx tanx = 1 / cos^2(x)

So, the derivative of the tangent function is 1 divided by the square of the cosine of x, expressed as 1 / cos^2(x).

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