a. what is the derivative of f(x)? b. what is...
a. what is the derivative of f(x)? b. what is the slope of the tangent line to the graph of f at the given x - values? f(x)=x^2 - 7x + 5; x = 5/2, x = 3 a. the derivative of f(x) is f(x)=
Answer
# Explanation:
## Step1: Apply power - rule
The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{2}-7x + 5$, the derivative of $x^{2}$ is $2x$ (since $n = 2$), the derivative of $-7x$ is $-7$ (since for $y=-7x^1$, $y^\prime=-7\times1\times x^{1 - 1}=-7$), and the derivative of the constant 5 is 0.
$f^\prime(x)=\frac{d}{dx}(x^{2})-\frac{d}{dx}(7x)+\frac{d}{dx}(5)$
$f^\prime(x)=2x-7+0$
$f^\prime(x)=2x - 7$
## Step2: Find slope at $x=\frac{5}{2}$
Substitute $x = \frac{5}{2}$ into $f^\prime(x)$.
$f^\prime(\frac{5}{2})=2\times\frac{5}{2}-7$
$f^\prime(\frac{5}{2})=5 - 7=-2$
## Step3: Find slope at $x = 3$
Substitute $x = 3$ into $f^\prime(x)$.
$f^\prime(3)=2\times3-7$
$f^\prime(3)=6 - 7=-1$
# Answer:
a. $2x - 7$
b. At $x=\frac{5}{2}$, slope is $-2$; at $x = 3$, slope is $-1$