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

In the realm of calculus, finding the derivative of a function is a fundamental operation. It allows us to understand the rate at which a function's output changes with respect to its input. This article delves into the process of finding the derivative of a specific function: $,f(x) = \left(\frac{\sin x}{x}\right)(3 x^2+5)$. We will explore the necessary calculus rules, step-by-step calculations, and provide insights to enhance your understanding of differentiation.

Understanding the Problem

Before diving into the solution, it's crucial to understand the structure of the function. Our function, $,f(x) = \left(\frac{\sin x}{x}\right)(3 x^2+5)$, is a product of two smaller functions. The first function is a quotient, $,\frac{\sin x}{x}$, and the second is a polynomial, $,3x^2 + 5$. This structure dictates that we will primarily use the product rule and the quotient rule of differentiation.

• Product Rule: The product rule states that the derivative of the product of two functions, $,u(x)$ and $,v(x)$, is given by: $,(uv)' = u'v + uv'$.
• Quotient Rule: The quotient rule states that the derivative of the quotient of two functions, $,u(x)$ and $,v(x)$, is given by: $,(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.

We will also need to recall the derivatives of basic functions:

• The derivative of $,\sin x$ is $,\cos x$.
• The derivative of $,x^n$ is $,nx^{n-1}$, where $,n$ is a constant. This is the power rule.
• The derivative of a constant is 0.

With these rules in mind, we can proceed to find the derivative of the given function.

Applying the Product Rule

Let's identify our two functions, $,u(x)$ and $,v(x)$, in the product: $,f(x) = u(x)v(x)$

• $$\,u(x) = \frac{\sin x}{x}$$

• $$\,v(x) = 3x^2 + 5$$

Now, we need to find the derivatives of $,u(x)$ and $,v(x)$ separately.

Finding u'(x): Applying the Quotient Rule

To find the derivative of $,u(x) = \frac{\sin x}{x}$, we apply the quotient rule. Let $,p(x) = \sin x$ and $,q(x) = x$. Then, $,u(x) = \frac{p(x)}{q(x)}$ and the quotient rule gives us:

$$\,u'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{[q(x)]^2}$$

We know that:

• $$\,p'(x) = \frac{d}{dx}(\sin x) = \cos x$$

• $$\,q'(x) = \frac{d}{dx}(x) = 1$$

Substituting these into the quotient rule formula, we get:

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

Finding v'(x): Applying the Power Rule

To find the derivative of $,v(x) = 3x^2 + 5$, we apply the power rule.

$$\,v'(x) = \frac{d}{dx}(3x^2 + 5)$$

Using the power rule and the fact that the derivative of a constant is zero, we get:

$$\,v'(x) = 3(2x) + 0 = 6x$$

Putting it Together: The Product Rule in Action

Now that we have $,u(x)$, $,v(x)$, $,u'(x)$, and $,v'(x)$, we can apply the product rule to find the derivative of $,f(x)$.

Recall the product rule: $,f'(x) = u'(x)v(x) + u(x)v'(x)$

Substituting our calculated derivatives, 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 of the function, but we can simplify it further.

Simplifying the Derivative

To simplify the derivative, we can first distribute the terms and then look for common factors to combine.

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

Expanding the numerator of the first term, we have:

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

Now, let's get a common denominator of $,x^2$ to combine the terms:

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

Combining like terms, specifically the $,\sin x$ terms, we get:

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

This is the simplified form of the derivative. We can also factor out an $,x$ from the first two terms in the numerator:

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

While this form is also correct, the previous form is generally considered more simplified.

The Final Answer

The derivative of the function $,f(x) = \left(\frac{\sin x}{x}\right)(3 x^2+5)$ is:

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

This expression represents the instantaneous rate of change of the function $,f(x)$ at any given point $,x$.

Visualizing the Derivative

To further understand the derivative, it's helpful to visualize the original function and its derivative. The derivative represents the slope of the tangent line to the original function at any point. Where the derivative is positive, the original function is increasing; where the derivative is negative, the original function is decreasing; and where the derivative is zero, the original function has a horizontal tangent (a local maximum or minimum).

Graphing the original function and its derivative can provide valuable insights into the behavior of the function. For example, we can identify intervals where the function is increasing or decreasing, and we can locate critical points (where the derivative is zero or undefined).

Common Mistakes and How to Avoid Them

When finding derivatives, several common mistakes can occur. Being aware of these pitfalls can help you avoid them.

• Incorrectly Applying the Product or Quotient Rule: Ensure you are applying the correct formulas and substituting the correct terms. Double-check your work, especially when dealing with complex functions.
• Forgetting the Chain Rule: If you have a composite function (a function within a function), remember to apply the chain rule. In our case, we didn't need the chain rule directly, but it's a crucial rule to remember for many differentiation problems.
• Errors in Simplification: After finding the derivative, simplifying the expression is important. However, be careful to avoid algebraic errors during simplification. Double-check your steps, especially when distributing terms or combining fractions.
• Incorrectly Differentiating Basic Functions: Make sure you know the derivatives of basic functions such as $,\sin x$, $,\cos x$, $,x^n$, etc. A mistake in the derivative of a basic function will propagate through the entire calculation.

Conclusion

Finding the derivative of a function like $,f(x) = \left(\frac{\sin x}{x}\right)(3 x^2+5)$ involves applying fundamental calculus rules such as the product rule and the quotient rule. By breaking down the function into smaller parts, finding the derivatives of those parts, and then combining them using the appropriate rules, we can successfully determine the derivative. Careful attention to detail, a solid understanding of the rules, and practice are key to mastering differentiation. This comprehensive guide provides a step-by-step solution and insights to enhance your understanding of this important calculus concept. Remember to always double-check your work and be mindful of common mistakes to ensure accuracy. The derivative, $,f'(x) = \frac{3x^3\cos x + 5x\cos x + 3x^2\sin x - 5\sin x}{x^2}$, allows us to analyze the behavior of the original function, including its rate of change and critical points. By visualizing the function and its derivative, we gain a deeper understanding of their relationship.