Questions: Differentiate the function. g(x)=5 x^-3(x^4-5 x^3+12 x-8) g'(x)=

Transcript text: Differentiate the function. \[ \begin{array}{l} g(x)=5 x^{-3}\left(x^{4}-5 x^{3}+12 x-8\right) \\ g^{\prime}(x)=\square \end{array} \]

Solution

Solution Steps

To differentiate the function \( g(x) = 5x^{-3}(x^4 - 5x^3 + 12x - 8) \), we will use the product rule and the power rule. The product rule states that if you have a function \( u(x)v(x) \), its derivative is \( u'(x)v(x) + u(x)v'(x) \). The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). First, identify \( u(x) = 5x^{-3} \) and \( v(x) = x^4 - 5x^3 + 12x - 8 \), then find their derivatives \( u'(x) \) and \( v'(x) \). Finally, apply the product rule to find \( g'(x) \).

Step 1: Identify the Functions and Their Derivatives

We have the function \( g(x) = 5x^{-3}(x^4 - 5x^3 + 12x - 8) \). We identify two functions:

\( u(x) = 5x^{-3} \)
\( v(x) = x^4 - 5x^3 + 12x - 8 \)

The derivatives are:

\( u'(x) = \frac{-15}{x^4} \)
\( v'(x) = 4x^3 - 15x^2 + 12 \)

Step 2: Apply the Product Rule

Using the product rule for differentiation, \( (uv)' = u'v + uv' \), we find: \[ g'(x) = \left(\frac{-15}{x^4}\right)(x^4 - 5x^3 + 12x - 8) + \left(5x^{-3}\right)(4x^3 - 15x^2 + 12) \]

Step 3: Simplify the Expression

Simplifying the expression for \( g'(x) \), we get: \[ g'(x) = \frac{5(4x^3 - 15x^2 + 12)}{x^3} - \frac{15(x^4 - 5x^3 + 12x - 8)}{x^4} \]

Further simplification yields: \[ g'(x) = 5 - \frac{120}{x^3} + \frac{120}{x^4} \]