differentiate the function. g(x)=4x^(-3)(x^4 - 4x^3 + 11x - 6) g(x)=

Explanation:

Step1: Expand the function

First, multiply $4x^{-3}$ with each term inside the parentheses: $g(x)=4x^{-3}\cdot x^{4}-4x^{-3}\cdot4x^{3}+4x^{-3}\cdot11x - 4x^{-3}\cdot6$ Using the rule $a^m\cdot a^n=a^{m + n}$, we get $g(x)=4x^{4-3}-16x^{-3 + 3}+44x^{-3+1}-24x^{-3}=4x-16 + 44x^{-2}-24x^{-3}$.

Step2: Differentiate term - by - term

The derivative of a constant is 0, and using the power rule $\frac{d}{dx}(x^n)=nx^{n - 1}$: The derivative of $4x$ is $4\times1\times x^{1-1}=4$. The derivative of $- 16$ is $0$. The derivative of $44x^{-2}$ is $44\times(-2)x^{-2-1}=-88x^{-3}$. The derivative of $-24x^{-3}$ is $-24\times(-3)x^{-3 - 1}=72x^{-4}$.

Step3: Combine the derivatives

$g'(x)=4-88x^{-3}+72x^{-4}$.

Answer:

$4-\frac{88}{x^{3}}+\frac{72}{x^{4}}$