d/dx[tanx]
To find the derivative of the function f(x) = tan(x), we use the rules of differentiation
To find the derivative of the function f(x) = tan(x), we use the rules of differentiation.
The derivative of the tangent function can be determined using the quotient rule or by expressing it in terms of sine and cosine functions.
Using the quotient rule:
For any function f(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, the derivative of f(x) is given by:
f'(x) = (u'(x)v(x) – u(x)v'(x)) / [v(x)]^2
Let u(x) = sin(x) and v(x) = cos(x), so f(x) = tan(x).
Now, let’s find the derivatives of u(x) and v(x):
u'(x) = cos(x) (derivative of sin(x) is cos(x))
v'(x) = -sin(x) (derivative of cos(x) is -sin(x))
Substituting the values into the quotient rule formula:
f'(x) = (cos(x)*cos(x) – sin(x)*(-sin(x))) / [cos(x)]^2
Simplifying the expression:
f'(x) = (cos^2(x) + sin^2(x)) / cos^2(x)
f'(x) = 1 / cos^2(x)
Now, we can further simplify the expression using the identity: cos^2(x) + sin^2(x) = 1
f'(x) = 1 / 1
f'(x) = 1
Therefore, the derivative of tan(x) is 1.
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