Question

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

Short answer

Question: Find the derivative of the function \(f(x) = x^\pi + \pi^x\). Answer: The derivative of the function is \(f'(x) = \pi \cdot x^{\pi - 1} + \pi^x \ln{\pi}\).

Step-by-step solution

Identify the terms in the sum

We will first compute the derivative of each term of the sum separately, then add the results together. 1. \(x^\pi\) 2. \(\pi^x\)

Use power rule for first term

Apply the power rule for derivatives to the first term, \(x^\pi\). $$\frac{d}{dx}(x^\pi) = \pi \cdot x^{\pi - 1}$$

Use logarithmic differentiation for second term

For the second term, \(\pi^x\), we must use logarithmic differentiation. First, let \(y = \pi^x\). Take the natural logarithm of both sides: $$\ln{y} = x \ln{\pi}$$ Now differentiate both sides of the equation with respect to \(x\): $$\frac{1}{y}\frac{dy}{dx} = \ln{\pi}$$ Multiplying through by \(y\) to isolate \(\frac{dy}{dx}\): $$\frac{dy}{dx} = y\ln{\pi}$$ Since \(y = \pi^x\), substitute to find the derivative of the second term: $$\frac{d}{dx}(\pi^x) = \pi^x \ln{\pi}$$

Combine the derivatives of both terms

Finally, add the derivatives of each term together to find the derivative of the given function: $$\frac{d}{dx}\left(x^{\pi}+\pi^{x}\right) = \pi \cdot x^{\pi - 1} + \pi^x \ln{\pi}$$

Key concepts

Exponential Derivative

In mathematics, understanding the exponential derivative is crucial when dealing with functions involving growth and decay. An exponential function typically has the form \(a^x\), where \(a\) is a constant, and \(x\) is the variable. The derivative of this type of function, particularly when \(a\) is a constant, is computed using a special rule in calculus called logarithmic differentiation. This rule allows us to express the derivative in terms of \(a^x\) and the natural logarithm of \(a\).
With logarithmic differentiation, the derivative of \(a^x\) (where \(a = \pi\), as demonstrated in our example) becomes \(a^x \ln(a)\). Here, since \(\pi^x = a^x\), the derivative is \(\pi^x \ln(\pi)\).
This method is particularly advantageous because it simplifies finding derivatives of exponential functions, especially those with bases other than the natural number \(e\). Understanding this process lays the groundwork for tackling more complex problems involving exponential functions.

Power Rule

The power rule is perhaps one of the most straightforward and versatile rules in differential calculus. It describes how to find the derivative of a power function, which is in the form \(x^n\), where \(n\) is any constant. This rule is succinctly given by:

• If \(y = x^n\), then the derivative \(\frac{dy}{dx} = nx^{n-1}\).

In our problem, \(x^\pi\) is an example where the power rule is effectively applied.
Here, \(n = \pi\). Applying the power rule gives \(\frac{d}{dx}(x^\pi) = \pi \cdot x^{\pi - 1}\). This process highlights the beauty and efficiency of the power rule.
Whenever dealing with exponents that are constants, the power rule is the go-to technique because of its simplicity and direct application, making it invaluable for differentiation tasks. It reduces the complexity of finding derivatives significantly, especially when dealing with polynomial-like expressions.

Natural Logarithm

The natural logarithm, denoted by \(\ln(x)\), is a critical mathematical function that arises frequently in calculus, particularly in logarithmic differentiation. It is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828.
When we take the derivative of an exponential function like \(a^x\), where \(a\) is a constant, the natural logarithm becomes useful. It aids in transforming the expression in a way that the power of \(x\) can be brought down, making differentiation manageable.

• For the function \(y = \pi^x\), taking the natural logarithm of both sides gives: \(\ln(y) = x \ln(\pi)\).
• Then, differentiating with respect to \(x\) using the implicit differentiation gives \(\frac{1}{y} \cdot \frac{dy}{dx} = \ln(\pi)\).

The natural logarithm simplifies the differentiation process and is particularly potent when used in combination with logarithmic differentiation strategies, especially for expressions where constants are raised to variable powers.