Math Problem Statement

Find the first and second derivatives of the function: y = (4x^3 + 7) / x

Solution

We are tasked with finding the first and second derivatives of the function:

y=4x3+7xy = \frac{4x^3 + 7}{x}

Step 1: Simplify the function

First, simplify the given function by dividing each term in the numerator by xx:

y=4x3x+7xy = \frac{4x^3}{x} + \frac{7}{x}

This simplifies to:

y=4x2+7x1y = 4x^2 + 7x^{-1}

Step 2: First derivative

To find the first derivative, yy', apply the power rule ddx[xn]=nxn1\frac{d}{dx}[x^n] = n x^{n-1} to each term:

[ y' = \frac{d}{dx} [4x^2] + \frac{d}{dx} [7x^{-1}] ]

Now, differentiate each term:

y=4(2x1)+7(1x2)y' = 4(2x^1) + 7(-1x^{-2})

Simplifying:

y=8x7x2y' = 8x - \frac{7}{x^2}

Step 3: Second derivative

To find the second derivative, yy'', differentiate y=8x7x2y' = 8x - \frac{7}{x^2}:

[ y'' = \frac{d}{dx} [8x] - \frac{d}{dx} \left[\frac{7}{x^2}\right] ]

For 8x8x, the derivative is 8.

For 7x2-\frac{7}{x^2}, use the power rule again:

ddx[7x2]=7(2x3)=14x3\frac{d}{dx} \left[-7x^{-2}\right] = -7(-2x^{-3}) = \frac{14}{x^3}

Thus, the second derivative is:

y=8+14x3y'' = 8 + \frac{14}{x^3}

Final results:

  • First derivative: y=8x7x2y' = 8x - \frac{7}{x^2}
  • Second derivative: y=8+14x3y'' = 8 + \frac{14}{x^3}

Would you like more details on any step, or have any further questions?

Here are 5 related questions to further expand:

  1. How would the derivatives change if the power of xx in the original function were higher or lower?
  2. Can you apply the product or quotient rule directly to this problem without simplifying first?
  3. How does the power rule for differentiation work when applied to negative exponents?
  4. What happens to the behavior of the function as xx approaches 0 or infinity?
  5. How would you interpret the second derivative physically or geometrically?

Tip: When simplifying functions before differentiating, breaking down terms can often help make the application of basic rules easier and less prone to error.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Power Rule
Simplifying Expressions

Formulas

Power rule: d/dx [x^n] = n * x^(n-1)
First and second derivatives

Theorems

Differentiation using Power Rule

Suitable Grade Level

High School - Grades 11-12