d/dx tanx
To find the derivative of tan(x) with respect to x, we will use the quotient rule
To find the derivative of tan(x) with respect to x, we will 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) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
In this case, g(x) = sin(x) and h(x) = cos(x). Therefore, we need to find the derivatives of sin(x) and cos(x), and then apply the quotient rule.
Derivative of sin(x):
The derivative of sin(x) with respect to x is cos(x). Therefore, g'(x) = cos(x).
Derivative of cos(x):
The derivative of cos(x) with respect to x is -sin(x). Therefore, h'(x) = -sin(x).
Now, we can use the quotient rule to find the derivative of tan(x):
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2
f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / (cos(x))^2
f'(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))
Now, using the trigonometric identity cos^2(x) + sin^2(x) = 1, we can simplify the expression further:
f'(x) = 1 / (cos^2(x))
Recall that tan(x) = sin(x) / cos(x), so we can rewrite the derivative as:
f'(x) = 1 / (cos^2(x)) = 1 / cos^2(x) = sec^2(x)
Therefore, the derivative of tan(x) with respect to x is sec^2(x).
In conclusion, d/dx tan(x) = sec^2(x).
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