Question

Evaluate the following derivatives. $$\frac{d}{d x}\left(x^{e}+e^{x}\right)$$

Short answer

Answer: The derivative of the function is \(ex^{e-1} + e^x\).

Step-by-step solution

Identify the individual terms to differentiate

Our given function is a sum of two terms: $$x^e$$ and $$e^x$$. We will find the derivative of each term separately and then add them up.

Differentiate $$x^e$$ with respect to x

The power rule states that if \(f(x) = x^n\) where \(n\) is a constant, then \(f'(x) = nx^{n-1}\). In our case, \(n = e\). Applying the power rule, we have: $$\frac{d}{d x}(x^e) = ex^{e-1}$$

Differentiate $$e^x$$ with respect to x

The derivative of the exponential function with respect to x is itself. Thus, we have: $$\frac{d}{d x}(e^x) = e^x$$

Combine the results

Now that we have the derivatives of each term, we need to add them up to find the derivative of the complete function. Therefore, $$\frac{d}{d x}\left(x^{e}+e^{x}\right) = ex^{e-1} + e^x$$ The derivative of the given function is $$ex^{e-1} + e^x$$.

Key concepts

Power Rule

The power rule is a fundamental tool in calculus that lets us differentiate functions of the form \(x^n\), where \(n\) is a constant. This rule states that if you have a function \(f(x) = x^n\), then its derivative \(f'(x) = nx^{n-1}\). The concept is simple: you bring the power down as a multiplication factor and then decrease the power by one. This is incredibly useful for polynomial differentiation.For example, if \(n = 3\), the derivative of \(x^3\) would be \(3x^2\). The power rule makes it easy to differentiate complex polynomial functions quickly. In our original exercise, \(x^e\), where \(e\) is a constant, is differentiated using the power rule to get \(ex^{e-1}\). The flexibility of the power rule makes it a fundamental part of calculus, allowing students to tackle a wide variety of problems with ease.

Exponential Function

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. A common example is \(e^x\), where \(e\) (approximately 2.718) is known as Euler's number, a crucial irrational number in mathematics. The fascinating property of this function is that its rate of change is proportional to its current value.When differentiating \(e^x\), the derivative is remarkably simple: it is the same \(e^x\). This makes exponential functions straightforward to work with in calculus because they simplify the process of finding derivatives. This property stems from the fact that exponential growth processes, common in nature and finance, follow this pattern.In our exercise, identifying \(e^x\) and finding its derivative showcases the ease of working with exponential functions due to their self-derivative nature.

Differentiation

Differentiation is a core concept in calculus that involves finding the derivative of a function, which represents the rate at which the function's value changes as its input changes. Derivatives are essential in understanding how functions behave and are used in diverse fields such as physics, engineering, and economics.To differentiate a function means to apply specific rules, such as the power rule or rules for exponential functions, to find this rate of change. Differentiation transforms a function into a derivative, symbolically represented as \(f'(x)\) or \(\frac{df}{dx}\).In our exercise, differentiation is used to find the derivative of the function \(x^e + e^x\). By applying the power rule to \(x^e\) and knowledge of derivatives of exponential functions to \(e^x\), we derived the rate of change for the entire function, summing these results as needed.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Its primary applications are in understanding change and motion, providing the tools needed to model and solve problems across science, engineering, and economics.This field is split into two major components: differentiation (finding derivatives) and integration (finding integrals). Differentiation allows us to determine how a function changes, while integration is concerned with the area under curves or accumulation of quantities.In solving the provided exercise, we used concepts from differential calculus, particularly the power rule and properties of exponential functions. These tools help us find the derivatives, which indicate how our function \(x^e + e^x\) changes with respect to \(x\). Calculus thus provides a framework for comprehensively analyzing and understanding a wide array of phenomena.