Question

Find the derivatives of the following functions:

(a) y = (sin²(x)) / (x₃ + 1).
(b) Suppose f(x) = 7x + 2cos(3x). Assuming the f⁻¹(x) exists, find (f⁻¹(x))₁.

Answer 1

To find the derivative of y = (sin²(x)) / (x₃ + 1), we apply the quotient rule and differentiate each part separately. For (f⁻¹(x))₁ in f(x) = 7x + 2cos(3x), we solve for y in terms of x and substitute it back into the equation.

Step-by-step explanation:

a) To find the derivative of y = (sin²(x)) / (x₃ + 1):

1. Using the quotient rule, we differentiate the numerator and denominator separately.
2. The derivative of sin²(x) is 2sin(x)cos(x) due to the chain and product rules.
3. The derivative of x₃ + 1 is 3x².
4. Now, we apply the quotient rule: (2sin(x)cos(x)(x₃ + 1) - sin²(x)(3x²)) / (x₃ + 1)².

b) Assuming f⁻¹(x) exists, to find (f⁻¹(x))₁ for f(x) = 7x + 2cos(3x):

1. Let y = 7x + 2cos(3x).
2. Switch x and y: x = 7y + 2cos(3y).
3. Solve for y in terms of x. This may require the use of trigonometric identities.
4. Substitute x back into the equation to find (f⁻¹(x))₁.