find the indicated derivative.\n\n\\( \\frac { d r } { d s } \\) if \\( r = s ^ { 3 } - 4 s ^ { 2 } + 1…

find the indicated derivative.\n\n\\( \\frac { d r } { d s } \\) if \\( r = s ^ { 3 } - 4 s ^ { 2 } + 1 \\)\n\n\\( \\frac { d r } { d s } = \\)

find the indicated derivative.\n\n\\( \\frac { d r } { d s } \\) if \\( r = s ^ { 3 } - 4 s ^ { 2 } + 1 \\)\n\n\\( \\frac { d r } { d s } = \\)

Answer

Explanation:

Step1: Differentiate each term

Differentiate (s^3) using the power rule (\frac{d}{ds}(s^n)=ns^{n - 1}), so (\frac{d}{ds}(s^3)=3s^{2}). Differentiate (-4s^2) using the power rule, (\frac{d}{ds}(-4s^2)=-4\times2s=-8s). Differentiate the constant term (1), (\frac{d}{ds}(1) = 0).

Step2: Combine the derivatives

By the sum - rule of differentiation (\frac{d}{ds}(u + v+w)=\frac{du}{ds}+\frac{dv}{ds}+\frac{dw}{ds}) (where (u = s^3), (v=-4s^2), (w = 1)), we have (\frac{dr}{ds}=\frac{d}{ds}(s^3-4s^2 + 1)=\frac{d}{ds}(s^3)-\frac{d}{ds}(4s^2)+\frac{d}{ds}(1)).

Answer:

(3s^{2}-8s)