Problem 40 - Calculus and its applications

Question

Find \(y^{\prime \prime}\) $$ \text { For } y=x^{7}-8 x^{2}+2, \text { find } d^{6} y / d x^{6} $$

Step-by-Step Solution

Answer

\( \frac{d^6 y}{dx^6} = 5040x \)

Step 1: Understand the Function

We are given the function \( y = x^7 - 8x^2 + 2 \) and asked to find the sixth derivative with respect to \( x \), denoted as \( \frac{d^6 y}{dx^6} \).

Step 2: Simplify the Calculation by Understanding Derivatives

Realize that deriving a polynomial involves decreasing the power of each term by one and multiplying by the original power, and that all polynomial terms of lower degree than the derivative order vanish. This means terms in a polynomial will eventually become zero after enough derivatives.

Step 3: Calculate the First Derivative

Calculate the first derivative: \( y' = \frac{d}{dx}(x^7) - \frac{d}{dx}(8x^2) + \frac{d}{dx}(2) = 7x^6 - 16x \).

Step 4: Calculate the Second Derivative

Compute the second derivative: \( y'' = \frac{d}{dx}(7x^6) - \frac{d}{dx}(16x) = 42x^5 - 16 \).

Step 5: Calculate the Third Derivative

Find the third derivative: \( y''' = \frac{d}{dx}(42x^5) = 210x^4 \).

Step 6: Calculate the Fourth Derivative

Calculate the fourth derivative: \( y^{(4)} = \frac{d}{dx}(210x^4) = 840x^3 \).

Step 7: Calculate the Fifth Derivative

Find the fifth derivative: \( y^{(5)} = \frac{d}{dx}(840x^3) = 2520x^2 \).

Step 8: Calculate the Sixth Derivative

Compute the sixth derivative: \( y^{(6)} = \frac{d}{dx}(2520x^2) = 5040x \).

Key Concepts

Polynomial Derivatives

Polynomial derivatives involve rules that help us find the rate of change of polynomial functions. This process works by applying the power rule. The power rule states that for a term in the form of \( ax^n \), the derivative is \( n \cdot ax^{n-1} \). This means:

• The exponent \( n \) becomes the coefficient.
• The exponent \( n \) is reduced by one.

For example, if you are given \( y = x^7 - 8x^2 + 2 \), each term is derived separately:

• The derivative of \( x^7 \) is \( 7x^6 \).
• The derivative of \( -8x^2 \) is \( -16x \).
• The derivative of a constant like 2 is zero.

Use this rule consistently to derive polynomial functions easily.

Higher Order Derivatives

Higher order derivatives refer to taking the derivative of a function multiple times. In other words, it involves repeatedly applying differentiation processes.

In this exercise, you are asked to determine the sixth derivative \( \frac{d^6 y}{dx^6} \). To do this, begin with the first derivative, then move to the second, and continue this process through to the sixth. The derivatives look like this:

• First derivative: \( y' = 7x^6 - 16x \)
• Second derivative: \( y'' = 42x^5 - 16 \)
• Third derivative: \( y''' = 210x^4 \)
• Fourth derivative: \( y^{(4)} = 840x^3 \)
• Fifth derivative: \( y^{(5)} = 2520x^2 \)
• Sixth derivative: \( y^{(6)} = 5040x \)

You'll notice that higher derivatives eliminate lower degree terms progressively, so by the sixth derivative, only a linear term is left. Understanding this pattern is key to working with polynomials and derivatives.

Differentiation Techniques

Differentiation techniques vary, but the simplest involves using the power rule, as applied in this exercise. Understanding how each differentiation step works will help in finding higher-order derivatives effectively.

Techniques include:

• Breaking down functions: Derive each term separately in polynomials.
• Seeing patterns: Recognize how each differentiated term influences the subsequent derivative.
• Applying rules often: Consistent application of the rules ensures no steps are missed.

Once you see that each differentiation process reduces the degree of the terms by one, it becomes manageable. This ongoing reduction simplifies the polynomial successively, highlighting why lower degree terms become zero. Efficiency in handling derivatives comes with practice, using these simple rules consistently.