let $f(x)=6x^{4}sqrt{x}+\frac{-2}{x^{2}sqrt{x}}$.\nhint: this derivative is easier to find if you first…

let $f(x)=6x^{4}sqrt{x}+\frac{-2}{x^{2}sqrt{x}}$.\nhint: this derivative is easier to find if you first change the terms with radicals to fractional exponents, then simplify using basic exponent rules.\n$f(x)=$\nquestion help: video message instructor\nsubmit question jump to answer

let $f(x)=6x^{4}sqrt{x}+\frac{-2}{x^{2}sqrt{x}}$.\nhint: this derivative is easier to find if you first change the terms with radicals to fractional exponents, then simplify using basic exponent rules.\n$f(x)=$\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Rewrite the function using fractional exponents

$$ \begin{align*} f(x)&=6x^{4}\cdot x^{\frac{1}{2}}+\frac{-2}{x^{2}\cdot x^{\frac{1}{2}}}\ &=6x^{4 + \frac{1}{2}}-2x^{-2-\frac{1}{2}}\ &=6x^{\frac{9}{2}}-2x^{-\frac{5}{2}} \end{align*} $$

Step2: Apply the power rule for differentiation

The power rule is ((x^{n})^\prime=nx^{n - 1}). For the first term (y = 6x^{\frac{9}{2}}), its derivative (y^\prime=6\times\frac{9}{2}x^{\frac{9}{2}-1}=27x^{\frac{7}{2}}). For the second term (y=-2x^{-\frac{5}{2}}), its derivative (y^\prime=-2\times(-\frac{5}{2})x^{-\frac{5}{2}-1}=5x^{-\frac{7}{2}}).

Answer:

(f^\prime(x)=27x^{\frac{7}{2}} + 5x^{-\frac{7}{2}})