Question

Differentiate the following funetions: \(u=\frac{a+b}{(a+b x)^{2}}\)

Short answer

#Solution# #tag_title#Final Answer#tag_content#The derivative of the function \(u=\frac{a+b}{(a+b x)^2}\) with respect to \(x\) is: $$\frac{d}{dx}\left( {\frac{{a+b}}{{(a+b x)^{2}}}} \right) = - \frac{(a+b)(2b(a+bx))}{((a+bx)^2)^2}$$

Step-by-step solution

Find v'(x)

Since the function \(v(x) = a + b\) is a constant function, its derivative with respect to x is: $$v'(x) = \frac{d}{dx}(a+b) = 0$$

Find w'(x)

The function \(w(x) = (a + bx)^2\) is a polynomial. In order to simplify the differentiation process, we can use the chain rule which states that: $$\frac{d}{dx}\left[ f\left( g\left( x \right) \right) \right] = f'\left( g\left( x \right) \right)g'\left( x \right)$$ Here we have outer function \(f(y) = y^2\) and inner function \(g(x) = a + bx\). First, we will find the derivative of both functions: $$f'(y) = \frac{d}{dy}(y^2) = 2y$$ $$g'(x) = \frac{d}{dx}(a + bx) = b$$ Now, we can plug these derivatives into the chain rule equation: $$w'(x) = f'(g(x))g'(x) = (2(a + bx))(b) = 2b(a + bx)$$

Apply Quotient Rule

Now that we have both \(v'(x)\) and \(w'(x)\), we can apply the quotient rule: $$\frac{d}{dx}\left(\frac{a+b}{(a+bx)^2}\right) = \frac{v'(x)w(x) - v(x)w'(x)}{(w(x))^2}$$ Plugging in the expressions and simplifying, we get: $$\frac{d}{dx}\left(\frac{a+b}{(a+bx)^2}\right) = \frac{(0)((a+bx)^2) - (a+b)(2b(a+bx))}{((a+bx)^2)^2}$$ $$= - \frac{(a+b)(2b(a+bx))}{((a+bx)^2)^2}$$

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative measures how a function changes as its input changes, essentially describing the function's rate of change.
In simpler terms, differentiation tells us how steep the function is at any given point.

• For example, in the original exercise, one function was a constant, leading to a derivative of zero.
• This happens because constant functions do not change as the input changes, meaning there is no slope.

Differentiation helps us understand many real-world phenomena, such as velocity, acceleration, and slopes of curves, by providing a way to calculate these rates of change precisely.
It is widely applied in physics, engineering, and economics, among other fields.

Quotient Rule

The quotient rule is a technique for differentiating functions that are divided by each other. This rule is particularly useful when you have a ratio of two differentiable functions and you need to find the derivative of the entire expression.The formula for the quotient rule is:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\]where

• \(u\) is the numerator function and \(v\) is the denominator function.
• \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively.

In the original problem, this rule was used to find the derivative of a function that was a quotient of a constant numerator and a polynomial denominator.
The key steps involved substituting the derivatives into the formula to simplify and compute the result.

Chain Rule

The chain rule is essential when differentiating a composite function, which is a function inside another function.It allows you to find the derivative of a composite function by differentiating the outer and inner functions separately and then multiplying the results.The chain rule formula is given by:\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]where:

• \(f\) is the outer function and \(g\) is the inner function.

In our example, the function involved \(f(y) = y^2\) and \(g(x) = a + bx\), leading to the application of the chain rule to find the derivative of \((a + bx)^2\).
This approach allows a complex differentiation task to be broken down into simpler, sequential parts.

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole-number exponents and multiplied by coefficients.They are sums of terms like \(ax^n\) where \(a\) is the coefficient and \(n\) is the non-negative integer exponent.
Common examples include linear functions, quadratic functions, and cubic functions.In terms of differentiation:

• The derivative of \(x^n\) is \(nx^{n-1}\).
• This is a straightforward rule that makes differentiating polynomial functions relatively simple.

For the given exercise, \(w(x) = (a + bx)^2\) is a polynomial, and differentiating it involved using both the chain rule and basic polynomial rules.
Understanding polynomial functions is crucial as they frequently appear in various mathematical contexts and real-life applications.