The Derivative of Tan(x) | Understanding the Chain Rule and Trigonometric Identity to Find the Solution

(d/dx) tan(x)

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. But first, let’s recall the definition of the tangent function:

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In trigonometric terms, for any angle x, the tangent function tan(x) is defined as the ratio of the sine of x to the cosine of x.

Now, let’s find the derivative of tan(x) with respect to x:

Using the quotient rule, for any two functions u(x) and v(x), the derivative of u(x)/v(x) is given by:

(d/dx) [u(x)/v(x)] = (v(x)(du/dx) – u(x)(dv/dx)) / v(x)^2

In this case, u(x) = sin(x) and v(x) = cos(x). So, we have:

(d/dx) tan(x) = (cos(x)(d/dx)(sin(x)) – sin(x)(d/dx)(cos(x))) / cos(x)^2

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

(d/dx) sin(x) = cos(x) (by differentiation of sin(x))
(d/dx) cos(x) = -sin(x) (by differentiation of cos(x))

Substituting these derivatives in the expression, we have:

(d/dx) tan(x) = (cos(x)(cos(x)) – sin(x)(-sin(x))) / cos(x)^2
= (cos^2(x) + sin^2(x)) / cos(x)^2

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

(d/dx) tan(x) = 1 / cos(x)^2

And, since tan(x) = sin(x) / cos(x), the final expression becomes:

(d/dx) tan(x) = 1 / cos(x)^2 = sec(x)^2

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

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