(d/dx) tan(x)
sec^2(x)
The derivative of tan(x) with respect to x can be found using the quotient rule.
tan(x) = sin(x) / cos(x)
Using the quotient rule:
(d/dx) tan(x) = [(cos(x) * d/dx(sin(x))) – (sin(x) * d/dx(cos(x)))] / cos^2(x)
Now, we need to find the derivatives of sin(x) and cos(x).
(d/dx) sin(x) = cos(x)
(d/dx) cos(x) = -sin(x)
Substituting these back into the quotient rule formula, we get:
(d/dx) tan(x) = [(cos(x) * cos(x)) – (sin(x) * -sin(x))] / cos^2(x)
Simplifying, we get:
(d/dx) tan(x) = (cos^2(x) + sin^2(x)) / cos^2(x)
Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we get:
(d/dx) tan(x) = 1 / cos^2(x)
Therefore, the derivative of tan(x) with respect to x is 1 / cos^2(x).
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