d/dx tan x
sec^2 x
To find the derivative of tan(x), we need to use the quotient rule:
tan(x) = sin(x) / cos(x)
Now, we can apply the quotient rule:
d/dx [sin(x)/cos(x)] = (cos(x) * d/dx[sin(x)] – sin(x) * d/dx[cos(x)]) / cos^2(x)
Using the product rule, we can find the derivatives of sin(x) and cos(x):
d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
Now, we can substitute these values back into the quotient rule:
d/dx [sin(x)/cos(x)] = (cos(x) * cos(x) – sin(x) * (-sin(x))) / cos^2(x)
Simplifying further, we get:
d/dx tan(x) = (cos^2(x) + sin^2(x)) / cos^2(x)
d/dx tan(x) = 1/cos^2(x)
Therefore, the derivative of tan(x) is 1/cos^2(x).
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