Question

Find \(d y / d x\). \(y=3 x^{5 / 3}+\sqrt{x}\)

Short answer

\(\frac{dy}{dx} = 5x^{\frac{2}{3}} + \frac{1}{2}x^{-rac{1}{2}}\).

Step-by-step solution

Identify the Function Components

The function given is composed of two terms: the first term is a polynomial with a fractional exponent, \(3x^{\frac{5}{3}}\), and the second term is a square root, equivalent to \(x^{\frac{1}{2}}\). Both terms can be differentiated using the rules of differentiation for powers of \(x\).

Apply Power Rule to First Term

Using the power rule, which states \(\frac{d}{dx} x^n = n \cdot x^{n-1}\), differentiate the term \(3x^{\frac{5}{3}}\). The derivative is \(\frac{d}{dx} [3x^{\frac{5}{3}}] = 3 \times \frac{5}{3} \cdot x^{\frac{5}{3} - 1} = 5x^{\frac{2}{3}}\).

Apply Power Rule to Second Term

Differentiate the second term, \(x^{\frac{1}{2}}\), using the power rule. The derivative is \(\frac{d}{dx} [x^{\frac{1}{2}}] = \frac{1}{2} \cdot x^{\frac{1}{2} - 1} = \frac{1}{2}x^{-rac{1}{2}}\).

Combine the Derivatives

Add together the derivatives of the individual terms to find \(\frac{dy}{dx}\). Therefore, \(\frac{dy}{dx} = 5x^{\frac{2}{3}} + \frac{1}{2}x^{-rac{1}{2}}\).

Key concepts

Power Rule

The power rule is a basic principle in calculus that simplifies the differentiation process, especially when dealing with polynomials and functions with exponents.
It helps us to find the derivative of a function without much hassle.
The power rule states that if you have a function in the form of \(x^n\), then its derivative is \(n \cdot x^{n-1}\).
In simple terms:

• Bring down the exponent as a coefficient in front of \(x\)
• Reduce the original exponent by 1

For example:
- To differentiate \(x^3\), apply the power rule to get \(3 \cdot x^{2}\).
It's important for students to understand this rule to handle more complex expressions involving variables raised to powers.
When dealing with coefficients and constants, as seen in our exercise, you simply multiply them by the derivative obtained. This makes the power rule efficient and effective for solving many initial calculus problems.

Fractional Exponents

Fractional exponents can look a bit confusing at first, but they actually make certain expressions easier to manage.
A fractional exponent like \(x^{\frac{5}{3}}\) can be thought of as \(\sqrt[3]{x^5}\).
This means raising x to the power of 5 and then taking the cube root.
Handling fractional exponents, especially when differentiating, is similar to dealing with whole number exponents:

• Apply the exponent as seen in the function
• Use the power rule as usual

In the given exercise, we had to differentiate terms with fractional exponents:
- For \(3x^{\frac{5}{3}}\), applying the power rule gave \(5x^{\frac{2}{3}}\).
Many calculus problems involve fractional exponents, so understanding how to work with them is key.
Simplifying them into radical form, while not necessary for differentiation, can often clarify their meaning.

Calculus Problem-Solving

Calculus problems often require breaking down a function into manageable chunks or components.
The goal is to systematically apply calculus principles like the power rule to solve a problem.
In our exercise, the function used was a combination of two terms: \(3x^{\frac{5}{3}}\) and \(\sqrt{x}\), recognized as \(x^{\frac{1}{2}}\).
Tackling calculus problems involves:

• Identifying each term within the function
• Determining the differentiation rule for each term
• Applying the rules appropriately
• Combining results where necessary

Sound problem-solving involves attention to detail and practice.
This ensures no step is skipped.
Using this structured approach helps students solve even the most complex calculus problems.

Derivative Calculation

Calculating the derivative provides insight into the rate of change of a function.
This is crucial in many real-world applications, such as physics and engineering.
In our exercise, we are required to find \( \frac{dy}{dx}\), the derivative of \( y \) with respect to \( x \).
The process involved:

• Recognizing each term in the given function as a point of focus
• Applying the power rule on each term separately
• Combining the derivatives of these terms to arrive at the overall derivative

For the given function, this meant differentiating both \(3x^{\frac{5}{3}}\) and \(x^{\frac{1}{2}}\) using the power rule.
The result of this derivative calculation was \(5x^{\frac{2}{3}} + \frac{1}{2}x^{-\frac{1}{2}}\).
Gaining confidence in derivative calculations forms the foundation of understanding changes, slopes, and more in calculus.