Finding the Second Derivative: Step-by-Step Solution

We are given two equations relating the variables $t$, $x$, and $y$:

• $t = e^{2x}$
• $y = \log_e(t^2)$

Our goal is to find the second derivative of $y$ with respect to $x$, which is denoted as $\frac{d^2y}{dx^2}$.

Simplifying the Expression for y

First, let's simplify the expression for $y$. We have $y = \log_e(t^2)$. Using the logarithm property $\log(a^b) = b \log(a)$, we can rewrite this as:

$\qquad y = 2 \log_e(t)$

Substituting t into the Equation for y

Now, we substitute the given expression for $t$, which is $t = e^{2x}$, into the simplified equation for $y$:

$\qquad y = 2 \log_e(e^{2x})$

Using the property $\log_e(e^a) = a$, we can simplify this further:

$\qquad y = 2 \times (2x)$

$\qquad y = 4x$

So, the relationship between $y$ and $x$ is simply $y = 4x$. This makes finding the derivatives much easier.

Calculating the First Derivative (dy/dx)

Now, we find the first derivative of $y$ with respect to $x$. We need to differentiate $y = 4x$ with respect to $x$:

$\qquad \frac{dy}{dx} = \frac{d}{dx}(4x)$

Using the constant multiple rule and the power rule for differentiation ($\frac{d}{dx}(x) = 1$):

$\qquad \frac{dy}{dx} = 4 \times \frac{d}{dx}(x)$

$\qquad \frac{dy}{dx} = 4 \times 1$

$\qquad \frac{dy}{dx} = 4$

The first derivative $\frac{dy}{dx}$ is a constant value, 4.

Calculating the Second Derivative (d²y/dx²)

Finally, we find the second derivative by differentiating the first derivative $\frac{dy}{dx}$ with respect to $x$. We need to differentiate the constant value 4 with respect to $x$:

$\qquad \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$

$\qquad \frac{d^2y}{dx^2} = \frac{d}{dx}(4)$

The derivative of any constant is always 0:

$\qquad \frac{d^2y}{dx^2} = 0$

Therefore, the second derivative of $y$ with respect to $x$ is 0.

Conclusion

Given $t = e^{2x}$ and $y = \log_e(t^2)$, we found that $y$ simplifies to $y = 4x$. The first derivative $\frac{dy}{dx}$ is 4, and the second derivative $\frac{d^2y}{dx^2}$ is 0.

Revision Table: Logarithm and Differentiation Rules

Additional Information: Understanding Second Derivatives

The second derivative $\frac{d^2y}{dx^2}$ represents the rate of change of the first derivative $\frac{dy}{dx}$. While the first derivative tells us about the slope of a function at a point (how fast $y$ is changing with respect to $x$), the second derivative tells us about the concavity of the function (how the slope is changing). If the second derivative is positive, the curve is concave up. If it's negative, the curve is concave down. If the second derivative is zero, as in this case, it means the rate of change of the slope is zero. For the function $y=4x$, which is a straight line, the slope is constant (always 4), so its rate of change is indeed 0.

This problem demonstrates how simplifying the function expression first can significantly ease the differentiation process. Recognizing the direct relationship between $y$ and $x$ after substitution and applying logarithm properties was key.