Question

find the derivative of the function. y = 10\sqrt{x}+8x^{\frac{1}{4}} \frac{dy}{dx}=\square

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$. So $y = 10x^{\frac{1}{2}}+8x^{\frac{1}{4}}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = ax^{n}$, then $\frac{dy}{dx}=anx^{n - 1}$.
For the first term $10x^{\frac{1}{2}}$, $a = 10$ and $n=\frac{1}{2}$. So its derivative is $10\times\frac{1}{2}x^{\frac{1}{2}-1}=5x^{-\frac{1}{2}}$.
For the second term $8x^{\frac{1}{4}}$, $a = 8$ and $n=\frac{1}{4}$. So its derivative is $8\times\frac{1}{4}x^{\frac{1}{4}-1}=2x^{-\frac{3}{4}}$.

Step3: Find the derivative of the whole function

Since $\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}$, where $u = 10x^{\frac{1}{2}}$ and $v = 8x^{\frac{1}{4}}$, we have $\frac{dy}{dx}=5x^{-\frac{1}{2}}+2x^{-\frac{3}{4}}$.

Answer:

$5x^{-\frac{1}{2}}+2x^{-\frac{3}{4}}$