Question

Differentiate the following functions: \(\frac{x^{6}+1}{x^{3}}+\)

Short answer

Question: Find the derivative of the following function: \(f(x) = \frac{x^6 + 1}{x^3}\). Answer: The derivative of the given function is: \(f'(x) = 3x^2 - 3x^{-4}\).

Step-by-step solution

Simplify the given function

Divide the two monomials to simplify the given function as follows: $$ f(x) = \frac{x^6 + 1}{x^3} = x^3 + \frac{1}{x^3}. $$

Apply the sum rule of differentiation

Differentiate each term of the simplified function using the sum rule: $$ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}\left(\frac{1}{x^3}\right). $$

Use the power rule of differentiation

Differentiate the first term using the power rule of differentiation: $$ \frac{d}{dx}(x^3) = 3x^2. $$

Rewrite the second term

Rewrite the second term using a negative exponent so it becomes a power of x: $$ \frac{1}{x^3} = x^{-3}. $$

Apply the power rule to the second term

Differentiate the second term using the power rule of differentiation: $$ \frac{d}{dx}(x^{-3}) = -3x^{-4}. $$

Combine the results

Combine the results of steps 3 and 5 using the sum rule: $$ f'(x) = 3x^2 - 3x^{-4}. $$ The derivative of the given function is: $$ f'(x) = 3x^2 - 3x^{-4}. $$

Key concepts

Power Rule

The power rule in calculus is a fundamental tool used for differentiation. It's a formula that simplifies finding the derivative of a power function. The rule expresses that if you have a function in the form \(x^n\), where \(n\) is any real number, the derivative of this function is \(nx^{n-1}\). Essentially, you multiply by the power and then subtract one from the exponent.

In our case, when differentiating \(x^3\), we apply this rule as follows:

• Identify the exponent, which is 3.
• Multiply the term by the exponent: \(3 \times x^3\).
• Subtract one from the exponent: \(3x^{3-1} = 3x^2\).

Using the power rule simplifies calculus by turning potentially complex algebraic expressions into very straightforward differentiation tasks.

Sum Rule

The sum rule is another cornerstone of calculus differentiation. This rule allows you to differentiate a sum of functions separately and then combine the results. It states that if you have a function \(f(x) = g(x) + h(x)\), the derivative \(f'(x)\) is simply the derivative of \(g(x)\) plus the derivative of \(h(x)\) together as \(f'(x) = g'(x) + h'(x)\).

For the function \(f(x) = x^3 + \frac{1}{x^3}\), after simplifying to \(x^3 + x^{-3}\) using negative exponents, we can apply the sum rule like this:

• Differentiate \(x^3\), which gives \(3x^2\).
• Differentiate \(x^{-3}\), which gives \(-3x^{-4}\).
• Add those derivatives: \(3x^2 + (-3x^{-4})\), resulting in \(3x^2 - 3x^{-4}\).

The sum rule is useful because it lets you handle complex functions piece by piece, ensuring each part is correctly differentiated.

Negative Exponents

Negative exponents in calculus offer a way to write expressions that involve division, which simplifies differentiation. The concept behind negative exponents is that they represent the reciprocal of the base raised to the positive exponent. So \(x^{-n}\) is the same as \(\frac{1}{x^n}\).

In the exercise, we turn \(\frac{1}{x^3}\) into \(x^{-3}\) to make applying the power rule easier. By rewriting it, we transform a division issue into a straightforward multiplication by the power and adjustment of the exponent.

• Recognize \(\frac{1}{x^3}\) as \(x^{-3}\).
• This makes applying the power rule possible and direct: \(-3x^{-4}\).

Understanding negative exponents is crucial as it converts functions to a form that is more compatible with standard differentiation rules.