question find the derivative of (y = x^{5sqrt{x}}). be sure to include parentheses around the arguments of…

Explanation:

Step1: Rewrite the function

Rewrite $y = x^{5\sqrt{x}}$ as $y=e^{\ln(x^{5\sqrt{x}})}$. Using the property $\ln(a^b)=b\ln(a)$, we have $y = e^{5\sqrt{x}\ln(x)}$. Let $u = 5\sqrt{x}\ln(x)$.

Step2: Differentiate $u$ using the product - rule

The product - rule states that if $u = f(x)g(x)$, then $u^\prime=f^\prime(x)g(x)+f(x)g^\prime(x)$. Here, $f(x)=5\sqrt{x}=5x^{\frac{1}{2}}$ and $g(x)=\ln(x)$. So $f^\prime(x)=\frac{5}{2}x^{-\frac{1}{2}}$ and $g^\prime(x)=\frac{1}{x}$. Then $u^\prime=\frac{5}{2}x^{-\frac{1}{2}}\ln(x)+5x^{\frac{1}{2}}\cdot\frac{1}{x}=\frac{5\ln(x)}{2\sqrt{x}}+\frac{5}{\sqrt{x}}=\frac{5(\ln(x) + 2)}{2\sqrt{x}}$.

Step3: Differentiate $y = e^u$ using the chain - rule

The chain - rule states that if $y = e^u$ and $u$ is a function of $x$, then $y^\prime=e^u\cdot u^\prime$. Since $y = e^{5\sqrt{x}\ln(x)}$ and $u^\prime=\frac{5(\ln(x)+2)}{2\sqrt{x}}$, we have $y^\prime=x^{5\sqrt{x}}\cdot\frac{5(\ln(x)+2)}{2\sqrt{x}}$.

Answer:

$x^{5\sqrt{x}}\cdot\frac{5(\ln(x)+2)}{2\sqrt{x}}$