Now, use implicit differentiation to differentiate y = (3√t)^t with respect to t, treating y as a differentiable function of t.

Steps to Solve

1. Identify the equation We start with the equation given:

$$ y = (3\sqrt{t})^t $$

2. Take the natural logarithm of both sides Using logarithmic differentiation, we take the natural log of both sides to simplify.

$$ \ln(y) = \ln((3\sqrt{t})^t) $$

Using the property of logarithms, we can bring down the exponent:

$$ \ln(y) = t \cdot \ln(3\sqrt{t}) $$

3. Differentiate both sides with respect to ( t ) Now we differentiate both sides. Remember that when differentiating ( y ), we apply the chain rule, giving us:

$$ \frac{1}{y} \frac{dy}{dt} = \frac{d}{dt}(t \cdot \ln(3\sqrt{t})) $$

4. Differentiate the right side We apply the product rule on the right side:

$$ \frac{d}{dt}(t \cdot \ln(3\sqrt{t})) = \frac{d}{dt}(t) \cdot \ln(3\sqrt{t}) + t \cdot \frac{d}{dt}(\ln(3\sqrt{t})) $$

The derivative of ( t ) is ( 1 ), and we need to find the derivative of ( \ln(3\sqrt{t}) ):

5. Differentiate ( \ln(3\sqrt{t}) ) Using the properties of logarithms:

$$ \ln(3\sqrt{t}) = \ln(3) + \frac{1}{2} \ln(t) $$

Thus,

$$ \frac{d}{dt}(\ln(3\sqrt{t})) = 0 + \frac{1}{2} \cdot \frac{1}{t} = \frac{1}{2t} $$

6. Combine results So, we have:

$$ \frac{1}{y} \frac{dy}{dt} = \ln(3\sqrt{t}) + t \cdot \frac{1}{2t} $$

This simplifies to:

$$ \frac{1}{y} \frac{dy}{dt} = \ln(3\sqrt{t}) + \frac{1}{2} $$

7. Solve for ( \frac{dy}{dt} ) Multiply both sides by ( y ):

$$ \frac{dy}{dt} = y \left( \ln(3\sqrt{t}) + \frac{1}{2} \right) $$

Finally, substitute back ( y = (3\sqrt{t})^t ):

$$ \frac{dy}{dt} = (3\sqrt{t})^t \left( \ln(3\sqrt{t}) + \frac{1}{2} \right) $$