Question

1. derivative: $f(x) = (2x^2 - 3)(x^3 + x + 1)$

1. derivative: $f(x) = \frac{600}{x + 4}$

Explanation:

Problem 9:

Step 1: Identify the product rule

We use the product rule: if \( f(x) = u(x)v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \). Let \( u(x) = 2x^2 - 3 \) and \( v(x) = x^3 + x + 1 \).

Step 2: Find \( u'(x) \)

Differentiate \( u(x) = 2x^2 - 3 \) with respect to \( x \). Using the power rule \( \frac{d}{dx}(x^n) = nx^{n - 1} \), we get \( u'(x) = 4x \).

Step 3: Find \( v'(x) \)

Differentiate \( v(x) = x^3 + x + 1 \) with respect to \( x \). Using the power rule, we have \( v'(x) = 3x^2 + 1 \).

Step 4: Apply the product rule

Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the product rule formula:
\[

$$\begin{align*} f'(x) &= u'(x)v(x) + u(x)v'(x) \\ &= 4x(x^3 + x + 1) + (2x^2 - 3)(3x^2 + 1) \end{align*}$$

\]

Step 5: Expand and simplify

First, expand \( 4x(x^3 + x + 1) = 4x^4 + 4x^2 + 4x \).
Then, expand \( (2x^2 - 3)(3x^2 + 1) = 6x^4 + 2x^2 - 9x^2 - 3 = 6x^4 - 7x^2 - 3 \).
Now, add the two expanded expressions together:
\[

$$\begin{align*} f'(x) &= (4x^4 + 4x^2 + 4x) + (6x^4 - 7x^2 - 3) \\ &= 4x^4 + 6x^4 + 4x^2 - 7x^2 + 4x - 3 \\ &= 10x^4 - 3x^2 + 4x - 3 \end{align*}$$

\]

Step 1: Rewrite the function

Rewrite \( f(x) = \frac{600}{x + 4} \) as \( f(x) = 600(x + 4)^{-1} \).

Step 2: Use the chain rule

The chain rule states that if \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x))h'(x) \). Let \( g(u) = 600u^{-1} \) and \( h(x) = x + 4 \).

Step 3: Find \( g'(u) \)

Differentiate \( g(u) = 600u^{-1} \) with respect to \( u \). Using the power rule \( \frac{d}{du}(u^n) = nu^{n - 1} \), we get \( g'(u) = -600u^{-2} \).

Step 4: Find \( h'(x) \)

Differentiate \( h(x) = x + 4 \) with respect to \( x \). We get \( h'(x) = 1 \).

Step 5: Apply the chain rule

Substitute \( u = h(x) = x + 4 \), \( g'(u) \), and \( h'(x) \) into the chain rule formula:
\[

$$\begin{align*} f'(x) &= g'(h(x))h'(x) \\ &= -600(x + 4)^{-2} \cdot 1 \\ &= -\frac{600}{(x + 4)^2} \end{align*}$$

\]

Answer:

\( f'(x) = 10x^4 - 3x^2 + 4x - 3 \)

Problem 10: