let y = f(x), where f(x) = 6x^3/2 + x^1/2. find the differential of the function. dy = (18x + 1)/(2√x) dx

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $y = 6x^{3/2}+x^{1/2}$, we differentiate each term separately.

Step2: Differentiate the first term

For $y_1 = 6x^{3/2}$, using the power - rule with $a = 6$ and $n=\frac{3}{2}$, we have $y_1^\prime=6\times\frac{3}{2}x^{\frac{3}{2}-1}=9x^{1/2}$.

Step3: Differentiate the second term

For $y_2 = x^{1/2}$, using the power - rule with $a = 1$ and $n=\frac{1}{2}$, we have $y_2^\prime=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-1/2}$.

Step4: Find the derivative of the function

The derivative of $y=f(x)$ is $y^\prime=f^\prime(x)=y_1^\prime + y_2^\prime=9x^{1/2}+\frac{1}{2}x^{-1/2}$.

Step5: Rewrite the derivative and find the differential

We rewrite $y^\prime$ as $y^\prime=\frac{18x + 1}{2\sqrt{x}}$. Since the differential $dy=f^\prime(x)dx$, we get $dy=\frac{18x + 1}{2\sqrt{x}}dx$.

Answer:

$dy=\frac{18x + 1}{2\sqrt{x}}dx$