d/dx tanx
To find the derivative of the function f(x) = tan(x), we use the quotient rule
To find the derivative of the function f(x) = tan(x), we use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative 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 g'(x) and h'(x) before we can apply the quotient rule.
The derivative of g(x) = sin(x) is g'(x) = cos(x). This is a basic trigonometric derivative.
Similarly, the derivative of h(x) = cos(x) is h'(x) = -sin(x). Again, this is a basic trigonometric derivative.
Now, we can apply the quotient rule to find the derivative of f(x) = tan(x):
f'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2
Expanding the numerator, we get:
f'(x) = (cos^2(x) + sin^2(x)) / [cos(x)]^2
Using the fundamental trigonometric identity sin^2(x) + cos^2(x) = 1, the numerator simplifies to:
f'(x) = 1 / [cos(x)]^2
Recall that tan(x) is defined as the ratio of sin(x) to cos(x). Using this definition, we can rewrite the derivative as:
f'(x) = 1 / [cos(x)]^2 = 1 / (1/[tan(x)])^2 = [tan(x)]^2
Therefore, the derivative of f(x) = tan(x) is f'(x) = [tan(x)]^2.
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