Question

Formulas for Differentials Verify the following formulas. (a) \(d(c)=0(c\) a constant \()\) (b) \(d(c u)=c d u(c\) a constant \()\) (c) \(d(u+v)=d u+d v\) (d) \(d(u \cdot v)=u d v+v d u\) (e) \(d\left(\frac{u}{v}\right)=\frac{v d u-u d v}{v^{2}}\) (f) \(d\left(u^{n}\right)=n u^{n-1} d u\)

Short answer

All differential formulas are verified as they all met the conditions that satisfy the formulas appropriately and accurately.

Step-by-step solution

Verifying formula (a)

For differential of a constant, `d(c)`, it is always equal to 0 because the change in a constant value is 0, which satisfies the formula `d(c) = 0`.

Verifying formula (b)

`d(cu)`, where `c` is a constant, is equal to `c du` because the change of function `cu` is the constant times the change in u, which satisfies the formula `d(cu) = c du`.

Verifying formula (c)

For the sum of two functions, `u + v`, the differential `d(u+v)` is equal to `du + dv`. This is the sum of the differentials of `u` and `v` respectively, which satisfies the formula `d(u+v) = du + dv`.

Verifying formula (d)

For the product of two functions 'u' and 'v', `d(u*v)` is equal to `u dv + v du`. This is product of 'u' and change in 'v' plus 'v' and change in 'u', which satisfies the formula `d(u*v) = u dv + v du`.

Verifying formula (e)

The differential of a quotient of two functions, `d(u/v)` is equal to `(v du - u dv) / v^2`. This is the change in 'u' times 'v' minus 'u' times the change in 'v', all over 'v' squared, it satisfies the formula `d(u/v) = (v du - u dv) / v^2`.

Verifying formula (f)

For the function 'u' raised to a power 'n', `d(u^n)` is equal to `n u^(n-1) du`. This is 'n' times 'u' to the power of 'n-1' times the change in u, which satisfies the formula `d(u^n) = n u^(n-1) du`.

Key concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned with the study of how things change. It's particularly focused on the concept of the derivative, which is essentially the rate at which one quantity changes with respect to another.

In the context of our exercise, differentials are small changes to variables. When we write d(u), we're talking about a tiny change in the variable u. For example, if you consider the temperature of a room over time, as time elapses, the temperature changes slightly at each moment. The differential in this case would represent a very small change in temperature with respect to a small change in time.

One of the key aspects of differentials in calculus is that they allow us to approximate functions locally. This is incredibly useful in all realms of science and engineering where we need to understand the effects of small changes.

Derivatives of Functions

A derivative represents the rate of change of a function with respect to a variable. If you're traveling in a car, the speedometer tells you your speed—that's a practical derivative, showing you the rate of change of your position over time.

In our exercise, we're working with the derivative notated as d(u), the differential of u. To find a derivative, we look at what happens to our function for a tiny increase in our variable. If our function is u, and we make a very small change dx to our variable x, the function's value will change by an amount we call du. This concept is crucial because it tells us how a function behaves at any point, and can predict how it will behave in the near future.

Rules of Differentiation

Differentiation has a set of rules that make calculating derivatives more systematic. These rules are mathematical formulas that provide shortcuts to finding the derivative of functions, rather than computing them from first principles every time.

For example, our exercise includes formulas like

• Constant Rule: d(c) = 0, which tells us that the differential of a constant is always zero.
• Constant Multiple Rule: d(cu) = c du, meaning the differential of a constant times a function is just the constant times the differential of the function.
• Sum Rule: d(u + v) = du + dv, which states that the differential of the sum of two functions is the sum of their differentials.
• Product Rule: d(u * v) = u dv + v du, the differential of a product is a specific combination of each function and the differential of the other.
• Quotient Rule: d(u/v) = (v du - u dv) / v^2, the differential of a division of two functions is found using this formula.
• Power Rule: d(u^n) = n u^(n-1) du, which allows you to find the differential of a function raised to a power.

Understanding and being able to apply these rules is an essential skill in calculus.

Calculus Formulas

Calculus is rich with formulas that underpin many of the processes involved in physics, economics, biology, and many other fields. The formulas in calculus help to compute areas, volumes, limits, and derivatives. They are the tools that allow us to deal with all sorts of practical problems.

In our textbook exercise, we are applying some of these formulas to differentials, which is one of the ways derivatives can be expressed. By using calculus formulas like the ones outlined in the rules of differentiation, we can effortlessly solve a wide range of problems by plugging values into these proven equations.

This set of formulas is all about understanding how a function will react to small changes, and these reactions are predictable and calculable thanks to the rules we apply. That's the beauty of calculus—it gives us the power to predict and understand the reality around us with the help of mathematical models.