91Ó°ÊÓ

When dealing with functions in the form of a fraction, it's essential to use the quotient rule for differentiation. The quotient rule provides a way to find the derivative of a quotient of two functions. If you have a function in the form \(\frac{u}{v}\), the quotient rule states: \[\frac{d}{dx} \bigg( \frac{u}{v} \bigg) = \frac{u'v - uv'}{v^2}\] Here, \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives. To apply the quotient rule, follow these steps:

• Differentiate \(u\) and denote it as \(u'\).
• Differentiate \(v\) and denote it as \(v'\).
• Substitute the values of \(u, u', v\), and \(v'\) into the quotient rule formula.
• Simplify the resulting expression.

In the given problem, \(u = 2\) and \(v = 7x^{2}+3x-1\). The derivative \(u' = 0\) because \(u\) is a constant. For \(v'\), we use the power rule, which we'll explore next.

The power rule is a fundamental tool in differentiation. It provides a straightforward way to differentiate terms of the form \(ax^n\). The rule states: \[\frac{d}{dx}(ax^n) = anx^{n-1}\] where \(a\) is a constant and \(n\) is the exponent. To apply the power rule, follow these steps:

• Identify the exponent \(n\) and the constant \(a\).
• Multiply \(a\) by \(n\).
• Reduce the exponent by 1.

Let's apply the power rule to each term in the denominator \(v = 7x^{2}+3x-1\). For \(7x^2\) the differentiation gives: \[ \frac{d}{dx}(7x^2) = 14x \] For \(3x\) the differentiation gives: \[ \frac{d}{dx}(3x) = 3 \] And the constant term \(-1\) differentiates to 0. Combining these results, we get \[v' = 14x + 3\].

Differentiation is a process in calculus used to find the rate of change of a function. It is one of the core concepts and techniques you will use in calculus. Here's a quick breakdown:

• It measures how a function changes as its input changes.
• The derivative is the function that describes this rate of change.
• Different rules apply based on the function's form (e.g., product rule, chain rule, power rule, and quotient rule).

In our problem, we combined two rules:

1. Used the power rule to differentiate the polynomial in the denominator.
2. Applied the quotient rule to find the derivative of the entire fraction.

The steps show that differentiation can involve multiple rules and techniques. First, we identified the function's form. Next, we found individual derivatives and combined them using the quotient rule formula.
Finally, we simplified our result to the final derivative:
\[ f'(x) = \frac{-28x - 6}{(7x^{2}+3x-1)^2}\] Remember, practice is key to mastering differentiation. By breaking problems into smaller steps, applying the correct rules, and simplifying, you will become more confident in solving these types of problems.