Question

Compute the derivative of the given function. $$f(x)=x^{\pi}+x^{1.9}+\pi^{1.9}$$

Short answer

The derivative of the function is \(\pi x^{\pi-1} + 1.9 x^{0.9}\).

Step-by-step solution

Differentiate the first term

The first term of the function is \(x^{\pi}\). Using the power rule \(\frac{d}{dx} x^n = n x^{n-1}\), we calculate the derivative: \(\pi \cdot x^{\pi - 1}\).

Differentiate the second term

The second term is \(x^{1.9}\). Again, apply the power rule: \(\frac{d}{dx} x^n = n x^{n-1}\). The derivative is \(1.9 \cdot x^{1.9 - 1} = 1.9 \cdot x^{0.9}\).

Differentiate the constant term

The last term, \(\pi^{1.9}\), is a constant with respect to \(x\). The derivative of any constant is 0.

Combine derivatives

Add the derivatives from each term to find the derivative of the entire function: \(\pi \cdot x^{\pi - 1} + 1.9 \cdot x^{0.9} + 0\).

Key concepts

Power Rule

The power rule is a fundamental concept in calculus for finding derivatives. When dealing with functions of the form \( x^n \), where \( n \) is any real number, the power rule simplifies the differentiation process. According to the power rule, the derivative of \( x^n \) is given by:

• \( \frac{d}{dx} x^n = n \cdot x^{n-1} \)

This means you bring the exponent down as a coefficient and then subtract one from the original exponent.
This concept is crucial when handling polynomial expressions, as it allows for quick computation. For instance, in differentiating \( x^\pi \) or \( x^{1.9} \), the power rule helps by transforming these into easier-to-handle terms.
Consistent practice with the power rule enhances understanding and proficiency in calculus, making it a favorite technique for students and educators alike.

Constant Rule

The constant rule in differentiation is straightforward yet essential. It states that the derivative of any constant is zero.
Regardless of the numeric value or how complex a constant appears, its derivative remains 0. This is because constants have no rate of change with respect to the variable of differentiation.
Take, for example, the term \( \pi^{1.9} \) in a function. Even though \( \pi^{1.9} \) may look intimidating due to its nature, it's treated simply as a constant in the context of differentiation. Thus, its derivative is:

• \( \frac{d}{dx} \pi^{1.9} = 0 \)

Remembering and applying the constant rule is fundamental not just in calculus but also in understanding how different components of a function behave during differentiation.

Differentiation

Differentiation is the mathematical process used to find the rate at which one quantity changes with respect to another. It is a key operation in calculus.
The primary objective of differentiation is to find the derivative, which is essentially the slope of the tangent line to the curve of a function at any given point.
This process involves applying various rules and principles, such as the power rule and constant rule, to different types of functions you'll encounter.
In the given exercise, differentiation is used to find the derivative of complex expressions involving variables raised to powers and constants.
This operation is critical in many fields, such as physics, engineering, and economics, where understanding changes and rates is necessary for analysis and modeling.

Mathematical Notation

Mathematical notation is the language of mathematics, providing symbols and conventions to express and solve mathematical problems. In calculus, standard notations are used to denote derivatives and differentiation processes.
The symbol \( \frac{d}{dx} \) signifies the derivative with respect to \( x \), while \( x^n \) represents a variable raised to power \( n \). These notations are crucial for clearly conveying mathematical ideas.
Additionally, an understanding of constants like \( \pi \), a widely recognized irrational number, is essential when differentiating functions involving such elements.
By mastering mathematical notation, students can efficiently interpret and manage calculus problems, making concepts clearer and solutions more accessible.
The effective use of notation not only aids in computation but also in communicating complex ideas succinctly within scientific and mathematical contexts.