Derivative Of (sin(x)/x)(3x^2 + 5) A Step-by-Step Solution

This article provides a step-by-step guide on how to find the derivative of the function f(x) = (sin(x)/x)(3x^2 + 5). This problem combines several important calculus concepts, including the product rule and the quotient rule, making it an excellent exercise for strengthening your understanding of differentiation. We will break down each step in detail, ensuring you grasp the underlying principles and can apply them to similar problems.

Understanding the Problem

The function we need to differentiate is:

$$f(x) = \left(\frac{\sin x}{x}\right)(3 x^2+5)$$

This function is a product of two expressions: (sin(x)/x) and (3x^2 + 5). Therefore, the first key concept we need to apply is the product rule. The product rule states that the derivative of two functions u(x) and v(x) multiplied together is given by:

$$(uv)' = u'v + uv'$$

In our case, we can define:

$$u(x) = \frac{\sin x}{x}$$

$$v(x) = 3x^2 + 5$$

So, our goal is to find u'(x) and v'(x), and then substitute them into the product rule formula. Finding u'(x) will require another rule: the quotient rule.

Applying the Quotient Rule

The quotient rule is used to find the derivative of a function that is a ratio of two other functions. It states that if we have a function u(x)/v(x), its derivative is:

$$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$$

For our u(x) = sin(x)/x, we can identify the numerator as a(x) = sin(x) and the denominator as b(x) = x. Thus, we need to find a'(x) and b'(x).

The derivative of a(x) = sin(x) is a fundamental derivative:

$$a'(x) = \cos x$$

The derivative of b(x) = x is also a basic derivative:

$$b'(x) = 1$$

Now, we can apply the quotient rule to find u'(x):

$$u'(x) = \frac{(\cos x)(x) - (\sin x)(1)}{x^2} = \frac{x \cos x - \sin x}{x^2}$$

In summary, applying the quotient rule to find the derivative of sin(x)/x involves identifying the numerator and denominator, finding their individual derivatives, and then substituting them into the quotient rule formula. This step is crucial because it provides one of the two key components needed for the product rule application later on. Understanding and correctly applying the quotient rule here is essential for successfully solving the overall problem. Remember to carefully track each term and sign to avoid common errors in differentiation.

Finding v'(x)

Now that we have u'(x), we need to find v'(x), where v(x) = 3x^2 + 5. This is a straightforward application of the power rule and the constant rule of differentiation. The power rule states that the derivative of x^n is nx^(n-1), and the constant rule states that the derivative of a constant is 0.

Applying the power rule to 3x^2, we get:

$$\frac{d}{dx}(3x^2) = 3 * 2x^(2-1) = 6x$$

The derivative of the constant term, 5, is 0:

$$\frac{d}{dx}(5) = 0$$

Therefore, v'(x) is the sum of these derivatives:

$$v'(x) = 6x + 0 = 6x$$

To reiterate, finding the derivative of v(x) = 3x^2 + 5 is a relatively simple process involving the power rule and the constant rule. This step highlights the importance of recognizing basic derivative rules and applying them correctly. The result, v'(x) = 6x, will be used in conjunction with u'(x) and the original functions u(x) and v(x) when we apply the product rule. This part of the problem reinforces the foundational aspects of differentiation, ensuring a solid understanding before moving on to more complex combinations of rules.

Applying the Product Rule

With u(x) = sin(x)/x, u'(x) = (x cos(x) - sin(x))/x^2, v(x) = 3x^2 + 5, and v'(x) = 6x in hand, we can now apply the product rule:

$$(uv)' = u'v + uv'$$

Substituting the expressions we found, we get:

$$f'(x) = \left(\frac{x \cos x - \sin x}{x^2}\right)(3x^2 + 5) + \left(\frac{\sin x}{x}\right)(6x)$$

This is the derivative, but we can simplify it further to obtain a cleaner expression.

Simplifying the Derivative

To simplify the derivative, we need to combine the two terms. First, let's multiply out the terms in the first part of the expression:

$$\left(\frac{x \cos x - \sin x}{x^2}\right)(3x^2 + 5) = \frac{(x \cos x - \sin x)(3x^2 + 5)}{x^2}$$

Now, let's expand the numerator:

$$(x \cos x - \sin x)(3x^2 + 5) = 3x^3 \cos x + 5x \cos x - 3x^2 \sin x - 5 \sin x$$

So, the first term becomes:

$$\frac{3x^3 \cos x + 5x \cos x - 3x^2 \sin x - 5 \sin x}{x^2}$$

The second term in our derivative is:

$$\left(\frac{\sin x}{x}\right)(6x) = 6 \sin x$$

Now, let's combine the two terms. To do this, we need a common denominator, which is x^2. So, we rewrite the second term:

$$6 \sin x = \frac{6x^2 \sin x}{x^2}$$

Now, we can add the two terms:

$$f'(x) = \frac{3x^3 \cos x + 5x \cos x - 3x^2 \sin x - 5 \sin x}{x^2} + \frac{6x^2 \sin x}{x^2}$$

Combine the numerators:

$$f'(x) = \frac{3x^3 \cos x + 5x \cos x - 3x^2 \sin x - 5 \sin x + 6x^2 \sin x}{x^2}$$

Simplify by combining like terms (the sin(x) terms):

$$f'(x) = \frac{3x^3 \cos x + 5x \cos x + 3x^2 \sin x - 5 \sin x}{x^2}$$

This is the simplified form of the derivative.

In summary, simplifying the derivative involves a series of algebraic manipulations. The key steps are expanding products, finding a common denominator, combining like terms, and factoring where possible. While the calculus aspect of the problem is complete once the product and quotient rules are applied, the simplification process is crucial for presenting the answer in a clear and concise form. This final simplified expression, f'(x) = (3x^3 cos(x) + 5x cos(x) + 3x^2 sin(x) - 5 sin(x))/x^2, represents the final solution to the problem.

Final Answer

Therefore, the derivative of the function f(x) = (sin(x)/x)(3x^2 + 5) is:

$$f'(x) = \frac{3x^3 \cos x + 5x \cos x + 3x^2 \sin x - 5 \sin x}{x^2}$$

This detailed walkthrough illustrates the application of the product rule and quotient rule in finding derivatives. By understanding each step and the underlying principles, you can confidently tackle similar problems in calculus.

Key Takeaways

• The product rule is essential for differentiating functions that are products of other functions.
• The quotient rule is necessary for differentiating functions that are ratios of other functions.
• Simplification is a crucial step in presenting the final answer in a clean and understandable format.
• Practice is key to mastering differentiation techniques. Work through various examples to build your skills and confidence.

This comprehensive guide should provide you with a solid understanding of how to find the derivative of (sin(x)/x)(3x^2 + 5). Remember to practice and apply these concepts to various problems to solidify your knowledge.