Question

Consider the function.4x - 59 (x) =(0,5)x² – 1'(a) Find the value of the derivative of the function at the given point.g'(0) =(b) Choose which differentiation rule(s) you used to find the derivative. (Select all that apply.)power rulequotient ruleproduct rule

Answer 1

In order to find the derivative for the division of two functions we need to apply the quotient rule,

$\begin{gathered} \frac{f(x)}{g(x)} \\ \frac{d}{dx}\lbrack\frac{f(x)}{g(x)}\rbrack=\frac{g(x)f^{\prime}(x)-f(x)g^{\prime}(x)}{(g(x))^2} \end{gathered}$

then, using the power rule of differentiation find the derivative of both numerator and denominator

$\begin{gathered} \frac{4x-5}{x^2-1} \\ derivative\text{ for the numerator: 4} \\ derivative\text{ for the denominator: 2x} \end{gathered}$

apply the quotient rule

$\begin{gathered} g^{\prime}(x)=\frac{(x^2-1)(4)-(4x-5)2x}{(x^2-1)^2} \\ g^{\prime}(x)=\frac{4x^2-4-8x^2+10x}{(x^2-1)^2} \\ g^{\prime}(x)=\frac{-4x^2+10x-4}{(x^2-1)^2} \end{gathered}$

evaluate the derivative on 0,

$\begin{gathered} g^{\prime}(0)=\frac{-4(0)^2+10(0)-4}{(0^2-1)^2} \\ simplify, \\ g^{\prime}(0)=\frac{-4}{(-1)^2} \\ g^{\prime}(0)=-4 \end{gathered}$

Answer:

a) g'(0)= -4

b) quotient rule and power rule