d tan x
sec^2 x
The expression d tan x is the derivative of the tangent function with respect to the variable x. We can find this derivative using the basic differentiation rules of calculus as follows:
First, we can write tangent as a ratio of sine and cosine:
tan x = sin x / cos x
Then, we can use the quotient rule to differentiate this expression:
(d/dx) tan x = (d/dx) [sin x / cos x]
= [cos x (d/dx) sin x – sin x (d/dx) cos x] / cos^2 x
Now, we need to find the derivatives of sine and cosine:
(d/dx) sin x = cos x
(d/dx) cos x = -sin x
Substituting these values into our expression, we get:
(d/dx) tan x = [cos x (cos x) – sin x (-sin x)] / cos^2 x
= (cos^2 x + sin^2 x) / cos^2 x
= 1 / cos^2 x
Therefore,
d tan x = 1 / cos^2 x
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