Question

question 2
if ( f(x)=3 x^{5}-4 x^{2}+3 x + 20 )
find the slope of the tangent line at ( x=-1 )

question 3
( f(x)=-4 x^{4}+2 x^{3}+2 )
what is the rate of change of ( f(x) ) at ( x=-1 )

Explanation:

Step1: Differentiate the function

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=3x^{5}-4x^{2}+3x + 20\), we have \(f^\prime(x)=3\times5x^{4}-4\times2x+3\times1+0\).
So, \(f^\prime(x)=15x^{4}-8x + 3\).

Step2: Substitute \(x = - 1\) into the derivative

Substitute \(x=-1\) into \(f^\prime(x)\):
\(f^\prime(-1)=15\times(-1)^{4}-8\times(-1)+3\).
First, \((-1)^{4}=1\), then \(15\times1 = 15\), \(-8\times(-1)=8\).
So, \(f^\prime(-1)=15 + 8+3\).

Answer:

\(26\)