y= (3x^3+2x)cotx
To find the derivative of the given function y = (3x^3 + 2x)cot(x), we can apply the product rule and chain rule of differentiation.
Step 1: Apply the product rule
Using the product rule, the derivative of (3x^3 + 2x)cot(x) with respect to x is calculated as follows:
dy/dx = [(3x^3 + 2x) * d(cot(x))/dx] + [cot(x) * d(3x^3 + 2x)/dx]
Step 2: Calculate the derivatives of cot(x), 3x^3, and 2x
To find the derivative of cot(x), we need to apply the chain rule. The derivative of cot(x) is -csc^2(x).
The derivative of 3x^3 with respect to x is obtained by multiplying the coefficient (3) by the power (3) and subtracting one from the exponent.
d(3x^3)/dx = 3 * 3x^2 = 9x^2
Similarly, the derivative of 2x with respect to x is simply 2.
Step 3: Substitute the derivatives into the equation from Step 1
Now, we substitute the calculated derivatives into the equation from Step 1:
dy/dx = [(3x^3 + 2x) * (-csc^2(x))] + [cot(x) * (9x^2 + 2)]
Simplifying further, the derivative of y is given by:
dy/dx = -csc^2(x)(3x^3 + 2x) + cot(x)(9x^2 + 2)
Hence, the derivative of y = (3x^3 + 2x)cot(x) with respect to x is -csc^2(x)(3x^3 + 2x) + cot(x)(9x^2 + 2).