Differentiate.
y = (4x)/9 − cot(x)
dy/dx = (4/9) + csc^2 x
note
d/dx (cos x/sin x )= [-sin^2 x - cos^2 x] /sin^2 x = -1/sin^2 x = -csc^2 x
Thank you for your help, but the website is telling me that answer is wrong.
Maybe you wrote it wrong. If you meant
y = (4x)/(9 − cot(x))
then
y' = [(4)(9-cotx)-(4x)(csc^2x)]/(9-cotx)^2
maybe rearranged a bit.
To differentiate the given function y = (4x)/9 − cot(x), we will apply the differentiation rules step by step.
First, let's differentiate the term (4x)/9 using the power rule. The power rule states that if we have a constant multiplied by x raised to the power of n, the derivative is obtained by multiplying the constant by the power of x raised to the power of (n-1). In this case, we have:
d/dx [(4x)/9] = (4/9) * d/dx (x) = (4/9) * 1 = 4/9
Next, we need to differentiate the term cot(x). Cot(x) is the reciprocal of the trigonometric function tan(x), which means cot(x) = 1/tan(x). To differentiate cot(x), we can apply the quotient rule.
The quotient rule states that if we have a fraction u(x)/v(x), the derivative is obtained by the formula:
d/dx [u(x)/v(x)] = [v(x) * du(x)/dx - u(x) * dv(x)/dx] / [v(x)]^2
In our case, u(x) = 1 and v(x) = tan(x). Let's differentiate these two terms separately.
1. Differentiating u(x) = 1: Since 1 is a constant, its derivative is 0. Therefore, du(x)/dx = 0.
2. Differentiating v(x) = tan(x): To differentiate tan(x), we can apply the chain rule, which states that if we have a composite function g(f(x)), the derivative is obtained by multiplying the derivative of the outer function g'(f(x)) with the derivative of the inner function f'(x).
In this case, g(x) = tan(x) and f(x) = x. The derivative of tan(x) is sec^2(x). Therefore, dv(x)/dx = sec^2(x).
Now, let's substitute these values back into the quotient rule formula:
d/dx [cot(x)] = [tan(x) * 0 - 1 * sec^2(x)] / [tan(x)]^2
= -sec^2(x) / tan^2(x)
To simplify this expression, we can use the trigonometric identity sec^2(x) = 1 + tan^2(x):
d/dx [cot(x)] = -sec^2(x) / tan^2(x)
= -1 / tan^2(x) - 1
= -1 - tan^2(x)
Now that we have the derivatives of both terms, we can combine them to find the derivative of the given function y = (4x)/9 - cot(x):
dy/dx = (4/9) - (-1 - tan^2(x))
= 4/9 + 1 + tan^2(x)
= (4 + 9 + 9tan^2(x))/9
= (13 + 9tan^2(x))/9
Therefore, the derivative of y = (4x)/9 - cot(x) is dy/dx = (13 + 9tan^2(x))/9.