Calculating the Derivative of Tan(x) Using the Quotient Rule and Chain Rule

d/dx(tanx)

To find the derivative of the function f(x) = tan(x), we can use the quotient rule or the chain rule

To find the derivative of the function f(x) = tan(x), we can use the quotient rule or the chain rule.

Applying the quotient rule, which states that the derivative of f(x) = g(x)/h(x) is given by [g'(x)h(x) – g(x)h'(x)] / h(x)^2, where g(x) = sin(x) and h(x) = cos(x), we have:

f'(x) = [d/dx(sin(x))cos(x) – sin(x)d/dx(cos(x))] / cos^2(x)

Now let’s calculate the derivatives of sin(x) and cos(x):

d/dx(sin(x)) = cos(x)

d/dx(cos(x)) = -sin(x)

Substituting these values into the expression for f'(x), we get:

f'(x) = [cos(x)cos(x) – sin(x)(-sin(x))] / cos^2(x)

Simplifying further:

f'(x) = [cos^2(x) + sin^2(x)] / cos^2(x)

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we have:

f'(x) = 1 / cos^2(x)

Finally, rewriting cos^2(x) as sec^2(x), we obtain the derivative of tan(x) as:

f'(x) = 1 / sec^2(x)

Alternatively, we can use the chain rule. The derivative of the function f(x) = tan(x) can be written as:

f'(x) = d/dx(tan(u)) * du/dx

Here, u = x, so du/dx = 1. The derivative of tan(u) with respect to u can be found using the basic derivative of tangent which is sec^2(u). Substituting these values:

f'(x) = sec^2(x) * 1

f'(x) = sec^2(x)

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

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