Question

Finding and Evaluating a Derivative In Exercises \(13-18,\) find \(f^{\prime}(x)\) and \(f^{\prime}(c) .\) $$ \text{Function} \quad \text {Value of} \quad c $$ $$ f(x)=\frac{\sin x}{x} \quad c=\frac{\pi}{6} $$

Short answer

The derivative of the function \(f(x)=\frac{\sin x}{x}\) is \(f^{\prime}(x)=\frac{x*\cos(x) - \sin(x)}{x^{2}}\). At \(x = \frac{\pi}{6}\), the derivative is \(f^{\prime}(c)=\frac{\frac{\pi}{6}*\cos(\frac{\pi}{6}) - \sin(\frac{\pi}{6})}{(\frac{\pi}{6})^{2}}\).

Step-by-step solution

Apply the Quotient Rule

The Quotient Rule states that the derivative of \(\frac{u}{v}\) is \(\frac{v*u^{\prime} - u*v^{\prime}}{v^{2}}\). Here, \(u = \sin(x)\) and \(v = x\). \nSo, \(u^{\prime} = \cos(x)\) and \(v^{\prime} = 1\). Substituting these into the Quotient rule gives us:\n \(f^{\prime}(x)=\frac{x*\cos(x) - \sin(x)*1}{x^{2}}\)

Simplify the Expression

The expression can be simplified to:\n \(f^{\prime}(x)=\frac{x*\cos(x) - \sin(x)}{x^{2}}\)

Substitute \(x=c\)

Substitute \(x\) with \(c = \frac{\pi}{6}\) to find the derivative at \(c\):\n \(f^{\prime}(c)=\frac{\frac{\pi}{6}*\cos(\frac{\pi}{6}) - \sin(\frac{\pi}{6})}{(\frac{\pi}{6})^{2}}\)

Key concepts

Quotient Rule

When it comes to calculus, differentiating complex functions is a fundamental skill, and the Quotient Rule is a powerful tool! It's specifically used when you have a function that is the division of two other functions. Here's how it rolls out: if you have a function in the form \(f(x) = \frac{u}{v}\), then its derivative \(f'(x)\) is given by the formula \[f'(x) = \frac{v\cdot u' - u\cdot v'}{v^2}\].
The key is to first identify your \(u\) and \(v\), which are the top (numerator) and bottom (denominator) parts of your fraction respectively, and then find their derivatives \(u'\) and \(v'\). After that, just plug into the formula, simplify, and voila— you've mastered the Quotient Rule! One pro tip: keep an eye on simplifying expressions; it can turn a cumbersome fraction into something much more manageable.

Derivative of sin(x)/x

The function \(\frac{\sin(x)}{x}\) can seem intimidating, but fear not—it's just a prime candidate for the Quotient Rule. With \(u = \sin(x)\) and \(v = x\), we calculate their derivatives: \(u' = \cos(x)\) (since the derivative of \(\sin\) is \(\cos\)), and \(v' = 1\) (because the derivative of \(x\) with respect to \(x\) is \(1\)). Apply the Quotient Rule to get \(f'(x)\) as shown in the solution. Notice how the challenge here isn't in crunching numbers but in understanding how to apply the rule. Remember, for every function like this, there's a smooth derivative waiting to be found—just follow the Quotient Rule steps closely.

Simplifying Expressions

After getting your derivative using the Quotient Rule, you might end up with a complicated expression. But we can't stop just yet—simplifying the expression is like cleaning up after a great meal; it's necessary and satisfying. Take the derivative \(\frac{x\cdot \cos(x) - \sin(x)}{x^2}\) for instance. Here's an opportunity to look carefully for common factors in the numerator and denominator, and also to reduce fractions to their lowest terms. In other cases, you might want to factor out common variables, or use trigonometric identities to make things neater. Simplifying might not change the derivative value, but it definitely helps to understand and work further with it. Plus, it's always nicer to look at a tidy mathematical expression!