Mastering Derivatives: How To Find The Derivative Of Tan X With Respect To X

d/dx(tanx)

sec²x

To find the derivative of tan x with respect to x, we can use the formula:

d/dx(tanx) = sec^2x

where sec x is the secant of x.

To derive this formula, we can use the definition of the tangent function:

tan x = sin x / cos x

Then we can apply the quotient rule:

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

Using the derivatives of sin x and cos x, which are cos x and -sin x respectively, we can simplify this expression as:

d/dx(tanx) = (cos x * cos x + sin x * sin x) / cos^2x

Using the trigonometric identity cos^2x + sin^2x = 1, we can simplify this expression as:

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

Since sec x is equal to 1 / cos x, we can rewrite the derivative of tangent as:

d/dx(tanx) = sec^2x

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

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