question find the derivative of y = 7x²√4{x}.

Answer:

$y'=14x\sqrt[4]{x}+\frac{7x^{2}}{4\sqrt[4]{x^{3}}}$

Explanation:

Step1: Rewrite the function

$y = 7x^{2}x^{\frac{1}{4}}=7x^{2 + \frac{1}{4}}=7x^{\frac{9}{4}}$

Step2: Apply power - rule

The power - rule for differentiation is $(x^{n})'=nx^{n - 1}$. $y'=7\times\frac{9}{4}x^{\frac{9}{4}-1}=\frac{63}{4}x^{\frac{5}{4}}$ Another way:

Step1: Use the product rule

The product rule is $(uv)' = u'v+uv'$, where $u = 7x^{2}$ and $v=\sqrt[4]{x}=x^{\frac{1}{4}}$. $u'=(7x^{2})' = 14x$ and $v'=(x^{\frac{1}{4}})'=\frac{1}{4}x^{-\frac{3}{4}}$

Step2: Calculate the derivative

$y'=u'v + uv'=14x\cdot x^{\frac{1}{4}}+7x^{2}\cdot\frac{1}{4}x^{-\frac{3}{4}}$ $y'=14x\sqrt[4]{x}+\frac{7x^{2}}{4\sqrt[4]{x^{3}}}$