Problem 36 - Single Variable Calculus

Question

Use logarithmic differentiation to find the derivative of the function. $$ y=x^{x^{2}} $$

Step-by-Step Solution

Answer

The derivative of \(y = x^{x^2}\) is \(\frac{dy}{dx} = x^{x^2}(2x\ln(x) + x)\).

Step 1: Take the natural logarithm of both sides of the equation

Taking the natural logarithm of both sides, we get: \[ \ln(y) = \ln(x^{x^2}) \]

Step 2: Simplify the equation using properties of logarithms

Utilizing the properties of logarithms, we get: \[ \ln(y) = x^2\ln(x) \]

Step 3: Differentiate both sides of the equation with respect to x

First, differentiate the left side of the equation with respect to x: \[ \frac{d(\ln(y))}{dx} = \frac{1}{y} \cdot \frac{dy}{dx} \] Next, differentiate the right side of the equation with respect to x: \[ \frac{d(x^2\ln(x))}{dx} = 2x\ln(x) + x^2\cdot\frac{1}{x} \] Now, we have: \[ \frac{1}{y} \cdot \frac{dy}{dx} = 2x\ln(x) + x \]

Step 4: Solve for dy/dx (y')

Multiplying both sides by y to isolate the derivative, we get: \[ \frac{dy}{dx} = y(2x\ln(x) + x) \] Since we know the original function y, we can substitute it back into the equation: \[ \frac{dy}{dx} = x^{x^2}(2x\ln(x) + x) \] This is the derivative of the given function with respect to x.

Key Concepts

Derivatives of Exponential Functions

When dealing with functions that have variables raised to the power of other variables, such as y = x^{x^2}, we're in the realm of exponential functions. The derivatives of these functions aren't always straightforward.
By definition, an exponential function is of the form y = a^{u(x)}, where a is a constant and u(x) is a function of x. The derivative of an exponential function with base e, also known as the natural exponential function, is simply the original function times the derivative of the exponent. However, when the base is not e, or the exponent is a function of x, we must use other techniques like logarithmic differentiation to find the derivative.

Properties of Logarithms

Logarithms are incredibly useful when dealing with exponential functions due to their unique properties that allow us to simplify complex expressions.
Key properties include:

• log_b(m * n) = log_b(m) + log_b(n): The logarithm of a product is the sum of the logarithms.
• log_b(m/n) = log_b(m) - log_b(n): The logarithm of a quotient is the difference of the logarithms.
• log_b(m^n) = n * log_b(m): The logarithm of a power is the exponent times the logarithm.

These properties allow us to rewrite the equations so that the differentiation rules can be applied more easily, as shown in our exercise by taking the logarithm of both sides and then simplifying.

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of another. It allows us to differentiate both sides of an equation with respect to a single variable. In the context of logarithmic differentiation, once we've taken the natural logarithm of both sides of our original function and applied logarithmic properties, we differentiate implicitly with respect to x.
This process involves differentiating each term while treating the other variable as a function of x, and then solving for the derivative of that variable. In our example, after applying the properties of logarithms, we differentiate both sides with respect to x, treating y as an implicit function of x. This helps us to find the derivative dy/dx.

Chain Rule in Calculus

The chain rule is a fundamental rule in calculus for finding the derivative of a composite function. It essentially says that if you have a function h(x) = f(g(x)), the derivative h'(x) is f'(g(x)) * g'(x). In layman's terms, it's the derivative of the outer function, with the inner function plugged in, multiplied by the derivative of the inner function.
In the context of logarithmic differentiation, after simplifying the function using logarithm properties, we often end up with a compound expression. Applying the chain rule allows us to differentiate these complex expressions with precision. As seen in the last differentiating step of the exercise, where we find the derivatives of x^2 * ln(x), the chain rule is critical in calculating the product of this derivative.