Question

Differentiate y = s - √s / s² with respect to s.

a) (1 - 2√s) / s³
b) (1 - √s) / s²
c) (2√s - 1) / s³
d) (√s - 1) / s²

Answer 1

After differentiating y = s - √ s / s² with respect to s, the correct answer is (a) (1 - 2√ s) / s³.

Step-by-step explanation:

To differentiate the function y = s - √s / s² with respect to s, we will apply the rules of differentiation for basic algebraic and rational functions.

Firstly, let's rewrite the function in a more differentiation-friendly form, separating it into two terms:

y = s / s² - (√s) / s²

Simplifying, we get:

y = 1/s - 1/(s√s)

Now let's differentiate each term separately using the power rule and the chain rule for differentiation:

For the first term (1/s), the derivative is:

-1 / s²

For the second term (-1/(s√s)), after rewriting as (-s^{-3/2}), the derivative is:

3/2 * s^{-5/2} = 3/(2s³√s)

Combining both derivatives, we have:

dy/ds = (-1 / s²) + (3 / (2s³√s)) = (3 - 2√s) / (2s³√s)

After further simplification by multiplying the numerator and the denominator by 2√s, the final derivative is:

dy/ds = (6 - 4√s) / (4s³) which simplifies to (3 - 2√s) / (2s³)

Therefore, the correct answer is (a) (1 - 2√s) / s³.