Question

attempt 1: 10 attempts remaining. find $f(a)$ for $f(x)=-2 + 8x-7x^{2}$. $f(a)=$

Explanation:

Step1: Recall power - rule of differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$, and the derivative of a constant $C$ is $0$, i.e., $(C)^\prime = 0$. For the function $f(x)=-2 + 8x-7x^{2}$, we differentiate each term separately.

Step2: Differentiate the constant term

The derivative of the constant term $- 2$ is $0$ since $\frac{d}{dx}(C)=0$ for a constant $C$. So, $\frac{d}{dx}(-2)=0$.

Step3: Differentiate the linear term

For the term $8x$, using the power - rule with $n = 1$, we have $\frac{d}{dx}(8x)=8\times1\times x^{1 - 1}=8$.

Step4: Differentiate the quadratic term

For the term $-7x^{2}$, using the power - rule with $n = 2$, we get $\frac{d}{dx}(-7x^{2})=-7\times2x^{2 - 1}=-14x$.

Step5: Find the derivative of the function

Combining the derivatives of each term, $f^\prime(x)=\frac{d}{dx}(-2)+\frac{d}{dx}(8x)+\frac{d}{dx}(-7x^{2})=0 + 8-14x=8 - 14x$.

Step6: Evaluate the derivative at $x = a$

To find $f^\prime(a)$, we substitute $x=a$ into $f^\prime(x)$. So, $f^\prime(a)=8-14a$.

Answer:

$8 - 14a$