Derivative of tan x
To find the derivative of tan x, we can use the quotient rule
To find the derivative of tan x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) can be found as:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
In the case of tan x, we can rewrite it as the quotient of sin x and cos x, since tan x is defined as sin x / cos x.
So, we have:
f(x) = sin x / cos x
Let’s find the derivatives of sin x and cos x first:
g(x) = sin x, so g'(x) = cos x
h(x) = cos x, so h'(x) = -sin x
Now, we can substitute these into the quotient rule formula:
f'(x) = (cos x * cos x – sin x * (-sin x)) / (cos x)^2
= (cos^2 x + sin^2 x) / (cos x)^2
Using the trigonometric identity sin^2 x + cos^2 x = 1, we can simplify this further:
f'(x) = 1 / (cos x)^2
Therefore, the derivative of tan x is 1 / (cos x)^2.
In summary:
The derivative of tan x is given by the expression 1 / (cos x)^2.
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