differentiate the expression with respect to $x$, assuming that $y$ is implicitly a function of…

differentiate the expression with respect to $x$, assuming that $y$ is implicitly a function of $x$.\n$sqrt{5x + y}$\n(express numbers in exact form. use symbolic notation and fractions where needed. write $y$ for $\frac{dy}{dx}$ if necessary.)\n$\frac{d}{dx}sqrt{5x + y}=$

differentiate the expression with respect to $x$, assuming that $y$ is implicitly a function of $x$.\n$sqrt{5x + y}$\n(express numbers in exact form. use symbolic notation and fractions where needed. write $y$ for $\frac{dy}{dx}$ if necessary.)\n$\frac{d}{dx}sqrt{5x + y}=$

Answer

Explanation:

Step1: Rewrite the function

Let $u = 5x + y$, then the function is $\sqrt{u}=u^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that $\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$. First, find the derivative of $u^{\frac{1}{2}}$ with respect to $u$: $\frac{d}{du}u^{\frac{1}{2}}=\frac{1}{2}u^{-\frac{1}{2}}$. Then find the derivative of $u = 5x + y$ with respect to $x$: $\frac{du}{dx}=5 + y'$.

Step3: Multiply the two derivatives

$\frac{d}{dx}\sqrt{5x + y}=\frac{1}{2}(5x + y)^{-\frac{1}{2}}\cdot(5 + y')=\frac{5 + y'}{2\sqrt{5x + y}}$.

Answer:

$\frac{5 + y'}{2\sqrt{5x + y}}$