Question

compute the derivatives of the following functions, where $k,k,a,a,n$ are positive constants: (a) the michaelis - menten kinetics function $v = \frac{kx}{k + x}$. $\frac{dv}{dx}=$ (b) the hill function $y = \frac{ax^{n}}{a^{n}+x^{n}}$. $\frac{dy}{dx}=$ note: you can earn partial credit on this problem. preview my answers submit answers you have attempted this problem 1 time. your overall recorded score is 50%. you have unlimited attempts remaining. page generated at 09/26/2025 at 12:05pm pdt webwork © 1996 - 2023 | theme math4 | ww_version 2.18 | pg_version 2.18 the webwork project

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$.

Step2: Find derivative of Michaelis - Menten function

For $v = \frac{Kx}{k + x}$, let $u = Kx$ and $v=k + x$. Then $u^\prime=K$ and $v^\prime = 1$.
By the quotient - rule, $\frac{dv}{dx}=\frac{K(k + x)-Kx\times1}{(k + x)^{2}}=\frac{Kk+Kx - Kx}{(k + x)^{2}}=\frac{Kk}{(k + x)^{2}}$.

Step3: Find derivative of Hill function

For $y=\frac{Ax^{n}}{a^{n}+x^{n}}$, let $u = Ax^{n}$ and $v=a^{n}+x^{n}$. Then $u^\prime=Anx^{n - 1}$ and $v^\prime=nx^{n - 1}$.
By the quotient - rule, $\frac{dy}{dx}=\frac{Anx^{n - 1}(a^{n}+x^{n})-Ax^{n}\times nx^{n - 1}}{(a^{n}+x^{n})^{2}}=\frac{Anx^{n - 1}a^{n}+Anx^{2n - 1}-Anx^{2n - 1}}{(a^{n}+x^{n})^{2}}=\frac{An a^{n}x^{n - 1}}{(a^{n}+x^{n})^{2}}$.

Answer:

(a) $\frac{Kk}{(k + x)^{2}}$
(b) $\frac{An a^{n}x^{n - 1}}{(a^{n}+x^{n})^{2}}$