## Calculus 8th Edition

$$\frac{dL}{dA}\bigg|_{A=12}=1.718inches/year.$$ Derivative of a function shows the rate of change of that function with respect to some quantity. Here, derivative of L, with respect to A, shows the rate of change of length of the Alaskan rockfish (in inches) with age (in years). Thus, dLdA=1.718inches/year at age, A=12 years is the rate of change of length at that particular age (A=12).
Length $L$ of Alaskan rockfish at age $A$ is given by, $$L(A)=0.0155A^3-0.372A^2+3.95A+1.21$$ Differentiating $L$ with respect to $A$ at $A=12$ years, $$\frac{dL}{dA}=3\times0.0155A^2-2\times0.372A+3.95A=0.0465A^2-0.744A+3.95$$ $$\frac{dL}{dA}\bigg|_{A=12}=0.0465(12)^2-0.744(12)+3.95=1.718inches/year.$$ Derivative of a function shows the rate of change of that function with respect to some quantity. Here, derivative of $L$, with respect to $A$, shows the rate of change of length of the Alaskan rockfish (in inches) with age (in years). Thus, $\frac{dL}{dA}=1.718$inches/year at age, $A=12$ years is the rate of change of length at that particular age ($A=12$).