Use logarithmic differentiation to find the derivative of the function.

y =sqrt(x)^3x

y = (√x)^3x

lny = 3x ln√x = 3/2 x lnx
1/y y' = 3/2 (lnx + x/x) = 3/2 (lnx + 1)
y' = 3/2 y (1+lnx)
= 3/2 (√x)^3x (1+lnx)
= 3/2 x^(3x/2) (1+lnx)

Why did the logarithm go to the party? Because it wanted to dance to the beat of differentiation! Now let's log our way to finding the derivative of this function.

First, let's take the natural logarithm of both sides of the equation, so we have:
ln(y) = ln(sqrt(x)^3x)

Now, we can apply some logarithmic magic to simplify the expression. Using the properties of logarithms, we can bring down the exponent:
ln(y) = 3x ln(sqrt(x))

Next step, let's differentiate both sides of the equation with respect to x. Leibniz is here to help us:
1/y * y' = (3x / 2sqrt(x)) + 3ln(sqrt(x))

Finally, let's solve for y':
y' = y * [(3x / 2sqrt(x)) + 3ln(sqrt(x))]

But wait, we need the value of y to get a numerical answer for y'. So, substitute the given value of x in the original function to find y. Once you know y, you can evaluate the expression for y' using that value.

To find the derivative of the function y = sqrt(x)^3x using logarithmic differentiation, we can follow these steps:

Step 1: Take the natural logarithm of both sides of the equation.
ln(y) = ln(sqrt(x)^3x)

Step 2: Use the properties of logarithms to simplify the expression.
ln(y) = ln(x^(3x))

Step 3: Apply the logarithm rules to bring down the exponent.
ln(y) = (3x) * ln(x)

Step 4: Differentiate both sides of the equation with respect to x.
(1/y) * y' = 3ln(x) + (3x/x)

Step 5: Simplify the expression.
y' = y * (3ln(x) + 3)

Step 6: Substitute the original function back into the derivative.
y' = sqrt(x)^3x * (3ln(x) + 3)

Therefore, the derivative of y = sqrt(x)^3x using logarithmic differentiation is y' = sqrt(x)^3x * (3ln(x) + 3).

To find the derivative of the function y = (sqrt(x))^3x, we can use logarithmic differentiation.

Step 1: Take the natural logarithm (ln) of both sides of the equation:
ln(y) = ln((sqrt(x))^3x)

Step 2: Apply properties of logarithms to simplify the expression:
ln(y) = 3x * ln(sqrt(x))

Step 3: Differentiate both sides of the equation with respect to x:
(d/dx) ln(y) = (d/dx) (3x * ln(sqrt(x)))

Step 4: Apply the chain rule on the right-hand side:
(d/dx) ln(y) = 3x * (d/dx) ln(sqrt(x)) + ln(sqrt(x)) * (d/dx) (3x)

Step 5: Differentiate ln(y) and ln(sqrt(x)):
(1/y) * (dy/dx) = 3x * (1/(2*sqrt(x))) * (1/2) * x^(-1/2) + ln(sqrt(x)) * 3

Step 6: Simplify the expression:
(dy/dx) / y = (3x/(2 * 2 * sqrt(x))) * (1/2) * x^(-1/2) + ln(sqrt(x)) * 3
(dy/dx) / y = (3x/(4 * sqrt(x))) * (1/2) * x^(-1/2) + ln(sqrt(x)) * 3

Step 7: Multiply both sides of the equation by y:
dy/dx = y * [(3x/(4 * sqrt(x))) * (1/2) * x^(-1/2) + ln(sqrt(x)) * 3]

Step 8: Substitute the original function y = (sqrt(x))^3x back into the equation:
dy/dx = (sqrt(x))^3x * [(3x/(4 * sqrt(x))) * (1/2) * x^(-1/2) + ln(sqrt(x)) * 3]

Simplifying further, we get the derivative of the function y = (sqrt(x))^3x as:
dy/dx = sqrt(x)^(3x - 1/2) * [3x/8 + 3ln(sqrt(x))]