Find the second derivative of the function h. h(x) = (10 – x2) (5x + 17)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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### Calculus Problem: Find the Second Derivative

Given:

\[ h(x) = (10 - x^2)(5x + 17) \]

we are asked to find the second derivative of the function \( h \).

First, determine the first derivative \( h'(x) \).

\[ h'(x) = \frac{d}{dx}[(10 - x^2)(5x + 17)] \]

To find \( h'(x) \), we need to use the product rule, which states:

\[ \frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v' \]

where \( u = (10 - x^2) \) and \( v = (5x + 17) \).

Next, find the first derivative of each function:

\[ u = 10 - x^2 \quad \Rightarrow \quad u' = -2x \]
\[ v = 5x + 17 \quad \Rightarrow \quad v' = 5 \]

Apply the product rule:

\[ h'(x) = (-2x)(5x + 17) + (10 - x^2)(5) \]
\[ h'(x) = -10x^2 - 34x + 50 - 5x^2 \]
\[ h'(x) = -15x^2 - 34x + 50 \]

Now, we find the second derivative \( h''(x) \), which is the derivative of \( h'(x) \):

\[ h''(x) = \frac{d}{dx}(-15x^2 - 34x + 50) \]
\[ h''(x) = -30x - 34 \]

Thus, the second derivative of the function \( h \) is:

\[ h''(x) = -30x - 34 \]
Transcribed Image Text:### Calculus Problem: Find the Second Derivative Given: \[ h(x) = (10 - x^2)(5x + 17) \] we are asked to find the second derivative of the function \( h \). First, determine the first derivative \( h'(x) \). \[ h'(x) = \frac{d}{dx}[(10 - x^2)(5x + 17)] \] To find \( h'(x) \), we need to use the product rule, which states: \[ \frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v' \] where \( u = (10 - x^2) \) and \( v = (5x + 17) \). Next, find the first derivative of each function: \[ u = 10 - x^2 \quad \Rightarrow \quad u' = -2x \] \[ v = 5x + 17 \quad \Rightarrow \quad v' = 5 \] Apply the product rule: \[ h'(x) = (-2x)(5x + 17) + (10 - x^2)(5) \] \[ h'(x) = -10x^2 - 34x + 50 - 5x^2 \] \[ h'(x) = -15x^2 - 34x + 50 \] Now, we find the second derivative \( h''(x) \), which is the derivative of \( h'(x) \): \[ h''(x) = \frac{d}{dx}(-15x^2 - 34x + 50) \] \[ h''(x) = -30x - 34 \] Thus, the second derivative of the function \( h \) is: \[ h''(x) = -30x - 34 \]
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