d/dx(cotx)
To find the derivative of cot(x) with respect to x, we can use the quotient rule
To find the derivative of cot(x) with respect to x, we can use the quotient rule. The cotangent function is defined as the reciprocal of the tangent function:
cot(x) = 1/tan(x).
Now, let’s find the derivative:
To apply the quotient rule, we have to differentiate the numerator and denominator separately.
First, we differentiate the numerator (1) which is a constant. The derivative of any constant is 0.
d/dx(1) = 0.
Next, we differentiate the denominator (tan(x)). The derivative of tan(x) with respect to x is sec^2(x) (the square of the secant function):
d/dx(tan(x)) = sec^2(x).
Now, using the quotient rule, we can write the derivative of cot(x) as:
d/dx(cot(x)) = (0 * tan(x) – 1 * sec^2(x)) / (tan^2(x)).
Since 0 * tan(x) = 0, the numerator simplifies to -sec^2(x).
We can further simplify the denominator:
tan^2(x) = (sin(x)/cos(x))^2 = sin^2(x)/cos^2(x).
Substituting these values back into the derivative equation, we have:
d/dx(cot(x)) = -sec^2(x) / (sin^2(x)/cos^2(x)).
To simplify this further, we can multiply the numerator and denominator by cos^2(x):
d/dx(cot(x)) = -sec^2(x) * (cos^2(x) / sin^2(x)).
Using the trigonometric identity sec^2(x) = 1 + tan^2(x), we have:
d/dx(cot(x)) = -(1 + tan^2(x)) * (cos^2(x) / sin^2(x)).
Expanding the numerator:
d/dx(cot(x)) = -((1 * cos^2(x) / sin^2(x)) + (tan^2(x) * cos^2(x) / sin^2(x))).
Simplifying the fraction in the numerator:
d/dx(cot(x)) = -(cos^2(x) / sin^2(x) + tan^2(x) * cos^2(x) / sin^2(x)).
Combining the two terms in the numerator:
d/dx(cot(x)) = -((1 + tan^2(x)) * cos^2(x) / sin^2(x)).
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the numerator:
d/dx(cot(x)) = -((1 + tan^2(x)) * cos^2(x) / (1 – cos^2(x))).
Using the identity 1 – cos^2(x) = sin^2(x), the denominator simplifies to:
d/dx(cot(x)) = -((1 + tan^2(x)) * cos^2(x) / sin^2(x)).
Finally, we notice that tan^2(x) + 1 = sec^2(x), so the numerator simplifies to:
d/dx(cot(x)) = -(sec^2(x) * cos^2(x) / sin^2(x)).
And since sec^2(x) = 1/cos^2(x), we can rewrite the equation as:
d/dx(cot(x)) = -(cos^2(x) / sin^2(x)) * (1 / cos^2(x)).
Simplifying further:
d/dx(cot(x)) = -(1 / sin^2(x)).
So, the derivative of cot(x) with respect to x is -1/sin^2(x).
Therefore,
d/dx(cot(x)) = -1/sin^2(x).
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