Question

Find the first and second derivative of the function.
G(r)=√r + (r)¹/⁶

Answer 1

The first derivative of G(r) = sqrt(r) + r^(1/6) is 1/2 * r^(-1/2) + 1/6 * r^(-5/6), and the second derivative is -1/4 * r^(-3/2) - 5/36 * r^(-11/6).

Step-by-step explanation:

To find the first and second derivative of the function G(r) = √r + r^{1/6}, we will use the rules of differentiation for each term individually.

The first derivative of G(r) with respect to r, denoted as G'(r), is found using the power rule and the chain rule for derivatives. Starting with the first term √r, which is r^{1/2}, the derivative is ½ r^{-1/2}. For the second term r^{1/6}, the derivative is ⅖ r^{-5/6}. So the first derivative is:

G'(r) = ½ r^{-1/2} + ⅖ r^{-5/6}

To find the second derivative, denoted as G''(r), we differentiate each term of the first derivative again.

For the term ½ r^{-1/2}, the derivative is -¼ r^{-3/2}. For the term ⅖ r^{-5/6}, the derivative is -⅖×6 r^{-11/6}.

Combining these, we get:

G''(r) = -¼ r^{-3/2} - ⅖×6 r^{-11/6}

Remember to always express the final derivatives in their simplest form.