d/dx tan(x)
To find the derivative of tan(x) with respect to x, we can use the quotient rule or the chain rule
To find the derivative of tan(x) with respect to x, we can use the quotient rule or the chain rule. Let’s use the quotient rule here.
The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative f'(x) is given by:
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 the sine function (sin(x)) and the cosine function (cos(x)). So, tan(x) = sin(x) / cos(x).
Let’s find the derivatives of sin(x) and cos(x) separately:
d/dx sin(x) = cos(x) (using the chain rule)
d/dx cos(x) = -sin(x) (using the chain rule)
Now, applying the quotient rule to tan(x), we have:
d/dx tan(x) = (cos(x)*cos(x) – sin(x)*(-sin(x))) / (cos(x))^2
Simplifying further:
d/dx tan(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))
Using the trigonometric identity cos^2(x) + sin^2(x) = 1:
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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