Problem 2 $$ \text { Find the } n \text ... [FREE SOLUTION]

Chapter 9: Problem 2

$$ \text { Find the } n \text {th derivative of } y=x^{3} e^{-x} \text { for } n>3 \text {. } $$

Short Answer

Expert verified

For $n>3$, the nth derivative of $y = x^3 e^{-x}$ is $y^{(n)} = e^{-x} \times P_n(x)$ where $P_n(x)$ is a polynomial.

Step by step solution

- Identify the function

The function given is $y = x^3 e^{-x}$.

- Use the product rule

To find the derivative of $y = x^3 e^{-x}$, use the product rule: $(uv)' = u'v + uv'$. Let $u = x^3$ and $v = e^{-x}$. We need to compute the derivatives $u' = 3x^2$ and $v' = -e^{-x}$.

- Compute the first derivative

$y' = (x^3)' e^{-x} + x^3 (e^{-x})'$. Substitute the derivatives: $y' = 3x^2 e^{-x} - x^3 e^{-x}$. Combine like terms: $y' = e^{-x} (3x^2 - x^3)$.

- Compute higher derivatives using the product rule

Notice that the derivative simplifies to a product of a polynomial and the exponential term. Increase powers and apply the product rule for each subsequent derivative. For example, the second derivative is $y'' = e^{-x}[( -3x^2 + 6x) + (3x^2 - x^3)]$. This simplifies further, revealing a pattern.

- Observe the pattern

For $n>3$, compute more derivatives noting the pattern: each derivative introduces more polynomial terms in decreasing powers of $x$ multiplied by $e^{-x}$. The coefficients are influenced by corresponding factorial terms.

- General formula for the nth derivative

The nth derivative for this general form $y = x^3 e^{-x}$ involves noticing that each differentiation introduces an additional factorial term and power reduction: $y^{(n)} = e^{-x} \times P_n(x)$ where $P_n(x)$ denotes a polynomial. For $n \ge 3$, higher derivatives adjust the coefficients in $P_n(x)$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule

To find the derivative of a function that is a product of two functions, we use the product rule. The product rule states that if you have a function in the form of $y = u(x) \times v(x)$, the first derivative $y'$ is given by: $(uv)' = u'v + uv'$

Here, $u = x^3$ and $v = e^{-x}$.
We need their individual derivatives: $u' = 3x^2$ and $v' = -e^{-x}$.

Using these derivatives, we apply the product rule: $y' = (x^3)' e^{-x} + x^3 (e^{-x})'$
Substitute the calculated derivatives to get: $y' = 3x^2 e^{-x} - x^3 e^{-x}$
Finally, factor out $e^{-x}$, giving us: $y' = e^{-x} (3x^2 - x^3)$.

Higher-Order Derivatives

Higher-order derivatives are derivatives of a function taken multiple times. For the function $y = x^3 e^{-x}$, we need to compute higher-order derivatives, particularly for $n > 3$.

Using the product rule repeatedly is essential. For each derivative, we apply the product rule to the resulting expression.
After the first derivative $y'$, the second derivative $y''$ can be computed similarly: $y'' = (e^{-x} (3x^2 - x^3))'$, which simplifies using the product rule again.

Notice a pattern in the polynomial part of the derivative: each higher-order derivative introduces smaller powers of $x$, and increasingly complex coefficients influenced by factorial terms of $n$. For example,

The 2nd derivative will involve terms of $x^2, x$, and a constant.

Therefore, for higher-order derivatives, we continuously simplify each result by applying the pattern.

Polynomial-Exponential Function

In our problem, the function $y = x^3 e^{-x}$ is a polynomial-exponential function. This means:

The function is a product of a polynomial $x^3$ and an exponential $e^{-x}$.
Each derivative consists of a polynomial multiplied by the exponential term $e^{-x}$.

Understanding the behaviour of polynomial-exponential functions helps in solving for higher-order derivatives.

With each differentiation, both the polynomial part $x^3$ and the exponential function $e^{-x}$ are affected.
The polynomial degree decreases by 1 with each differentiation, while the exponential part remains the same.

The general nth derivative takes the form: $y^{(n)} = e^{-x} \times P_n(x)$, where $P_n(x)$ is a polynomial derived from the original polynomial. The coefficients and terms of $P_n(x)$ become more intricate due to the cumulative effect of differentiation, often including factorial components reflecting the order of the derivative.

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91影视

$$ \text { Find the } n \text {th derivative of } y=x^{3} e^{-x} \text { for } n>3 \text {. } $$

Short Answer

Step by step solution

- Identify the function

- Use the product rule

- Compute the first derivative

- Compute higher derivatives using the product rule

- Observe the pattern

- General formula for the nth derivative

Key Concepts

Product Rule

Higher-Order Derivatives

Polynomial-Exponential Function

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