Answer:
The slope of the graph at x=-1 is 2
Step-by-step explanation:
Instant rate of change
Given a real function f(x), the instant rate of change with respect to x is defined as the derivative of f which coincides with the instant value of the slope of the tangent line in the point (x,y).
We have the function:
[tex]\displaystyle f(x) = x^5+\frac{1}{x^3}+7[/tex]
Prepare the function to apply the power rule of the derivative:
[tex]\displaystyle f(x) = x^5+x^{-3}+7[/tex]
Recall the power rule:
[tex](x^n)' = nx^{n-1}[/tex]
Also, the derivative of a constant is zero.
Taking the derivative:
[tex]\displaystyle f'(x) = 5x^4-3x^{-4}[/tex]
Evaluating for x=-1:
[tex]\displaystyle f'(-1) = 5(-1)^4-3(-1)^{-4}[/tex]
[tex]\displaystyle f'(-1) = 5*1-3*1=2[/tex]
The slope of the graph at x=-1 is 2
