Question

Find the derivative of the algebraic function. R(s) = (s^5 + 5)^4

Answer 1

The derivative of R(s) = (s^5 + 5)^4 is found using the chain rule and results in R'(s) = 20s^4(s^5 + 5)^3.

Step-by-step explanation:

The student asks for the derivative of the algebraic function R(s) = (s^5 + 5)^4. To find this, we'll use the chain rule of differentiation. The chain rule states that if you have a composite function g(f(x)), then the derivative g'(f(x)) is g'(f(x)) * f'(x). In this case, our outer function is g(u) = u^4 (with u being the inner function s^5 + 5), and our inner function is f(s) = s^5 + 5. Applying the chain rule:

1. Differentiate the outer function with respect to the inner function: g'(u) = 4u^3.
2. Differentiate the inner function with respect to s: f'(s) = 5s^4.

Combining these results, the derivative of R(s) is:

R'(s) = g'(u) * f'(s) = 4(s^5 + 5)^3 * 5s^4 = 20s^4(s^5 + 5)^3.

Answer 2

To find the derivative of the algebraic function R(s) = (s^5 + 5)^4, use the chain rule.

Step-by-step explanation:

In order to find the derivative of the algebraic function R(s) = (s^5 + 5)^4, we can use the chain rule. By applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function.

The derivative of (s^5 + 5)^4 is 4(s^5 + 5)^3 times the derivative of the inner function, which is the derivative of s^5 + 5. The derivative of s^5 + 5 is 5s^4. Therefore, the derivative of R(s) = (s^5 + 5)^4 is 4(s^5 + 5)^3 * 5s^4.