Question

If \(T=\frac{\pi \mu \Omega h D^{3}}{4 c}\) find \(\frac{\partial T}{\partial D}\) and \(\frac{\partial T}{\partial c}\).

Short answer

Answer: \(\frac{\partial T}{\partial D} = \frac{3 \pi \mu \Omega h D^2}{4 c}\) and \(\frac{\partial T}{\partial c} = \frac{-\pi \mu \Omega h D^3}{16 c^2}\).

Step-by-step solution

Find \(\frac{\partial T}{\partial D}\)

To find the partial derivative of the function \(T\) with respect to \(D\), we will treat all other variables as constants and use the rules of differentiation. Using the Power Rule on the \(D^3\) term, we have: \(\frac{\partial}{\partial D}(D^3) = 3D^2\) Now, we will differentiate \(T\) with respect to \(D\). Since the only term containing \(D\) is the \(D^3\) term, the partial derivative is: \(\frac{\partial T}{\partial D} = \frac{\pi \mu \Omega h (3D^2)}{4 c} = \frac{3 \pi \mu \Omega h D^2}{4 c}\)

Find \(\frac{\partial T}{\partial c}\)

Now, we will find the partial derivative of \(T\) with respect to \(c\). To do this, we will treat all other variables as constants and use the rules of differentiation. We can apply the Quotient Rule to the function \(T\): \(\frac{\partial T}{\partial c} = \frac{0 - \frac{\partial c}{\partial c}(\pi \mu \Omega h D^3)}{(4 c)^2}\) Using the Constant Rule on the \(\frac{\partial c}{\partial c}\) term and simplifying, we get: \(\frac{\partial T}{\partial c} = \frac{-\pi \mu \Omega h D^3}{16 c^2}\) So the partial derivatives of \(T\) with respect to \(D\) and \(c\) are: \(\frac{\partial T}{\partial D} = \frac{3 \pi \mu \Omega h D^2}{4 c}\) and \(\frac{\partial T}{\partial c} = \frac{-\pi \mu \Omega h D^3}{16 c^2}\)

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus, often referred to as the process of finding a derivative. In the context of functions of several variables, partial differentiation is the act of isolating one variable to determine how the function changes as that single variable changes while keeping all other variables constant. This is akin to observing how the terrain's slope varies in a single direction when standing on a hill, disregarding slopes in all other directions.

When we write \( \frac{\partial f}{\partial x} \), for example, we're expressing the rate at which the function \( f \) changes with respect to the variable \( x \), assuming all other variables in \( f \) do not vary. This granularity allows for precise control and understanding in multidisciplinary fields like physics, economics, and engineering where functions frequently depend on multiple factors.

Power Rule

The power rule is a quick and efficient way to find the derivative of a function where the variable is raised to a power, represented generally as \( x^n \). According to the power rule, the derivative of \( x^n \) with respect to \( x \) is \( n \cdot x^{n-1} \).

In our exercise, the power rule was applied to differentiate \( D^3 \) with respect to \( D \) giving us \( 3D^2 \). This rule greatly simplifies computations and is one of the first techniques taught when learning how to differentiate functions. Understanding and applying the power rule is essential, as it frequently appears in various mathematical problems, including those with partial derivatives.

Quotient Rule

The quotient rule is another derivative tool, used when a function is divided by another. This is expressed as \( f(x) = \frac{g(x)}{h(x)} \), where both \( g(x) \) and \( h(x) \) are themselves functions of \( x \). The quotient rule dictates that the derivative of \( f(x) \) with respect to \( x \) is \( \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2} \).

Understanding this rule is crucial when facing complex functions involving divisions, such as \( \frac{\pi \mu \Omega h D^{3}}{4 c} \) in the given exercise. It allows us to determine how fast the ratio of two functions changes with respect to an input variable.

Mathematical Constants

Mathematical constants are specific numbers that have fixed values and occur frequently across various areas of mathematics and physics. Pi (\( \pi \) ), for instance, is a constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. Constants such as \( \pi \) hold constant values and do not change when differentiating with respect to a variable.

In the process of differentiation, constants such as \( \mu \) and \( \Omega \) in our exercise are treated as constants and do not alter the outcome of the derivative unless they are part of the variable term that is being differentiated. It's essential to recognize these constants to properly apply the rules of differentiation.