The Chain Rule of Differentiation: Finding the Derivative of tan(x) with respect to x

d/dx(tanx)

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

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

Recall that the chain rule states that if y = f(g(x)), then the derivative of y with respect to x is given by the product of the derivative of f with respect to g and the derivative of g with respect to x.

In this case, let’s assign f(u) = tan(u) and g(x) = x. So, y = f(g(x)) = tan(x).

Now, let’s calculate the derivatives:

The derivative of f(u) = tan(u) with respect to u, denoted as f'(u), is given by sec^2(u). (This is a standard trigonometric derivative.)

The derivative of g(x) = x with respect to x is simply 1.

So, applying the chain rule, we have:

dy/dx = f'(g(x)) * g'(x)

dy/dx = sec^2(x) * 1

dy/dx = sec^2(x)

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

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