Question

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}(2 x)^{2 x}$$.

Short answer

Answer: The derivative of the function \((2x)^{2x}\) with respect to \(x\) is \((2x)^{2x}(2\ln(2x) + 2)\).

Step-by-step solution

Take the natural logarithm of the function

To use logarithmic differentiation, we first take the natural logarithm (ln) of both sides of the function: $$\ln y = \ln((2x)^{2x})$$ where \(y = (2x)^{2x}\).

Simplify the equation using log properties

We can simplify the equation using the power rule of logarithms, which states that \(\ln(a^b) = b\ln(a)\): $$\ln y = 2x \ln(2x)$$

Differentiate both sides with respect to \(x\)

Now we differentiate both sides of the equation with respect to \(x\) using implicit differentiation: $$\frac{dy}{dx}\cdot\frac{1}{y} = \frac{d}{dx}(2x \ln(2x))$$

Apply the product rule on the right-hand side of the equation

To differentiate the right side of the equation, we need to apply the product rule, which states that \((uv)' = u'v + uv'\). Let \(u = 2x\) and \(v = \ln(2x)\): $$\frac{1}{y}\cdot\frac{dy}{dx} = (2)(\ln(2x)) + (2x)(\frac{1}{2x})(2)$$

Simplify the equation

Simplify the equation: $$\frac{1}{y}\cdot\frac{dy}{dx} = 2\ln(2x) + 2$$

Solve for the derivative, \(\frac{dy}{dx}\)

Multiply both sides of the equation by \(y = (2x)^{2x}\) to obtain the derivative, \(\frac{dy}{dx}\): $$\frac{dy}{dx} = (2x)^{2x}(2\ln(2x) + 2)$$ In conclusion, the derivative of the given function with respect to \(x\) is: $$\frac{d}{dx}(2x)^{2x} = (2x)^{2x}(2\ln(2x) + 2)$$

Key concepts

Derivative Calculation

To find the derivative of a complex function, like \((2x)^{2x}\), you can use a special technique called logarithmic differentiation. Normally, figuring out derivatives involves straightforward rules, but some functions, especially those involving both products and powers, need more effort. Logarithmic differentiation simplifies this process by using the natural logarithm (\(\ln\)) to break the function into simpler parts.
This method helps us easily apply differentiation rules and eventually solve for the derivative.Here’s a general process to remember:

• Take the natural logarithm of both sides of the equation where applicable.
• Simplify the expression using logarithmic properties.
• Differentiate implicitly using the usual rules.
• Solve for the derivative.

Understanding each of these steps is key to mastering derivative calculations for more complex functions.

Product Rule

When differentiating expressions like \(2x \ln(2x)\), which involve the product of two functions, the product rule is your go-to tool. The product rule states that if you have a function \(u(x)\) times another function \(v(x)\), then the derivative can be calculated as:\[(uv)' = u'v + uv'\]So, break down the product into parts to simplify the differentiation process:
1. Let \(u = 2x\) and \(v = \ln(2x)\).
2. Differentiate each part separately: \(u' = 2\) and \(v' = \frac{1}{2x}(2)\).
3. Apply the product rule formula: \( (2)(\ln(2x)) + (2x)(\frac{1}{2x})(2)\).
This completes the differentiation of the product, turning a seemingly complicated expression into a manageable calculation.

Exponential Functions

Exponential functions are a common type of function that grows rapidly, typically in the form \(a^{x}\) or \((bx)^{cx}\). In this exercise, \((2x)^{2x}\) represents a base that changes with \(x\), which is a specific form of an exponential function involving a parameter and variable as both the base and exponent.
Handling such functions often requires logarithmic differentiation, as directly applying basic differentiation isn't feasible.
By taking the natural logarithm, we transform the expression into one that is easier to differentiate.Key points to remember:

• Understand the form and behavior of exponential functions.
• Use logarithmic properties to convert them into simpler forms.
• Apply appropriate differentiation techniques to handle them efficiently.

This understanding is crucial when solving real-world problems involving exponential growth or decay.

Logarithmic Properties

Logarithmic properties are crucial when simplifying expressions before differentiation. They allow us to break down complex functions, like \((2x)^{2x}\), into simpler parts using known logarithmic identities. The key property used here is:- Power rule: \(\ln(a^{b}) = b \ln(a)\)This property simplifies the expression \(\ln((2x)^{2x})\) to \(2x \ln(2x)\). Once in this form, you can more easily apply differentiation rules.
Other useful properties include:

• \(\ln(1) = 0\)
• \(\ln(a \times b) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)

These rules are indispensable tools in transforming logarithmic or exponential equations into solvable problems, giving you the foundation to apply other differentiation techniques effectively.