Understanding how to find the derivative of the tangent function and its relationship with the secant function

derivative of tan x

To find the derivative of the tangent function, we can use the trigonometric identity:

1 + tan^2(x) = sec^2(x)

Now, differentiating both sides with respect to x:

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

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

Using the chain rule on the left side:

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

Simplifying further:

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

Dividing through by 2 * tan(x):

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

Therefore, the derivative of the tangent function, tan(x), is equal to the secant function, sec(x)

To find the derivative of the tangent function, we can use the trigonometric identity:

1 + tan^2(x) = sec^2(x)

Now, differentiating both sides with respect to x:

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

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

Using the chain rule on the left side:

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

Simplifying further:

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

Dividing through by 2 * tan(x):

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

Therefore, the derivative of the tangent function, tan(x), is equal to the secant function, sec(x).

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