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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