d/dx tan x
sec^2 x
To find the derivative of tan(x) with respect to x, we use the quotient rule. Recall that the quotient rule states:
(f/g)’ = (f’g – fg’) / g^2
where f’ and g’ are the derivatives of f and g, respectively.
In this case, f = sin(x) and g = cos(x), so we have:
tan(x) = sin(x) / cos(x)
Taking the derivatives of f and g, we get:
f’ = cos(x) and g’ = -sin(x)
Using the quotient rule, we have:
(tan(x))’ = ((cos(x))(cos(x)) – (sin(x))(-sin(x))) / (cos(x))^2
Simplifying, we have:
(tan(x))’ = (cos^2(x) + sin^2(x)) / (cos^2(x))
Since cos^2(x) + sin^2(x) = 1, we have:
(tan(x))’ = 1 / cos^2(x)
And we know that:
sec^2(x) = 1 / cos^2(x)
So we can write:
(tan(x))’ = sec^2(x)
Therefore, the derivative of tan(x) with respect to x is sec^2(x).
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