Questions: Use the Product Rule or Quotient Rule to find the derivative. f(x)=(-5x^3-4)/(7x^4-10) f'(x)=

Transcript text: Use the Product Rule or Quotient Rule to find the derivative. \[ f(x)=\frac{-5 x^{3}-4}{7 x^{4}-10} \] \[ f^{\prime}(x)= \]

Solution

Solution Steps

To find the derivative of the function \( f(x) = \frac{-5x^3 - 4}{7x^4 - 10} \), we will use the Quotient Rule. The Quotient Rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then its derivative is given by:

\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]

Here, \( g(x) = -5x^3 - 4 \) and \( h(x) = 7x^4 - 10 \). We need to find \( g'(x) \) and \( h'(x) \), and then apply the Quotient Rule formula.

Step 1: Define the Functions \( g(x) \) and \( h(x) \)

Given the function \( f(x) = \frac{-5x^3 - 4}{7x^4 - 10} \), we identify: \[ g(x) = -5x^3 - 4 \] \[ h(x) = 7x^4 - 10 \]

Step 2: Compute the Derivatives \( g'(x) \) and \( h'(x) \)

We find the derivatives of \( g(x) \) and \( h(x) \): \[ g'(x) = -15x^2 \] \[ h'(x) = 28x^3 \]

Step 3: Apply the Quotient Rule

Using the Quotient Rule: \[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \] Substitute \( g(x) \), \( h(x) \), \( g'(x) \), and \( h'(x) \): \[ f'(x) = \frac{(-15x^2)(7x^4 - 10) - (-5x^3 - 4)(28x^3)}{(7x^4 - 10)^2} \]

Step 4: Simplify the Expression

Simplify the numerator: \[ f'(x) = \frac{-105x^6 + 150x^2 + 140x^6 + 112x}{(7x^4 - 10)^2} \] Combine like terms: \[ f'(x) = \frac{35x^6 + 112x + 150x^2}{(7x^4 - 10)^2} \]