a. find the derivative of f at x. that is, fi...
a. find the derivative of f at x. that is, find f(x) b. find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function. f(x)=x² - 4, x = - 3,x = - 1 a. f(x)= (simplify your answer. use integers or fractions for any numbers in the expression.)
Answer
# Explanation:
## Step1: Apply power - rule for differentiation
The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For the function $f(x)=x^{2}-4$, the derivative of $x^{2}$ is $2x$ (using the power - rule with $n = 2$) and the derivative of a constant ($-4$) is $0$. So, $f^\prime(x)=\frac{d}{dx}(x^{2}-4)=\frac{d}{dx}(x^{2})-\frac{d}{dx}(4)$.
$f^\prime(x)=2x+0 = 2x$
## Step2: Find the slope at $x=-3$
Substitute $x = - 3$ into $f^\prime(x)$.
$f^\prime(-3)=2\times(-3)=-6$
## Step3: Find the slope at $x=-1$
Substitute $x=-1$ into $f^\prime(x)$.
$f^\prime(-1)=2\times(-1)=-2$
# Answer:
a. $f^\prime(x)=2x$
b. When $x = - 3$, the slope is $-6$; when $x=-1$, the slope is $-2$