consider the following function: f(s, t) = s³ + 14s³t⁴ + 3st³ + 19 step 1 of 2: find ∂f/∂s. answer ∂f/∂s =

Explanation:

Step1: Differentiate term - by - term

When differentiating $f(s,t)=s^{3}+14s^{3}t^{4}+3st^{3}+19$ with respect to $s$, treat $t$ as a constant. For the first term $s^{3}$, using the power rule $\frac{d}{ds}(s^{n})=ns^{n - 1}$, we have $\frac{d}{ds}(s^{3}) = 3s^{2}$. For the second term $14s^{3}t^{4}$, since $t^{4}$ is a constant, $\frac{d}{ds}(14s^{3}t^{4})=14t^{4}\frac{d}{ds}(s^{3})=14t^{4}\times3s^{2}=42s^{2}t^{4}$. For the third term $3st^{3}$, since $t^{3}$ is a constant, $\frac{d}{ds}(3st^{3})=3t^{3}\frac{d}{ds}(s)=3t^{3}$. For the fourth term $19$, since it is a constant, $\frac{d}{ds}(19) = 0$.

Step2: Sum up the derivatives of each term

$\frac{\partial f}{\partial s}=3s^{2}+42s^{2}t^{4}+3t^{3}$.

Answer:

$3s^{2}+42s^{2}t^{4}+3t^{3}$