Question

For$f\left(x\right)=-3x^2+5x-2,\mathrm{find}\frac{f(x+h)-f\left(x\right)}{h},h\ne 0$

Short answer

The required answer is$-6x-3h+5$

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=-3x^2+5x-2$

Step 2. Calculation

Substitute $x+h\mathrm{for}x$in the given function and expand the expression.

$$\begin{gathered} f(x+h)=-3{(x+h)}^2+5(x+h)+2 \\ =-3(x^2+2xh+h^2)+5x+5h+2 \\ =-3x^2-6xh-3h^2+5x+5h+2 \end{gathered}$$

Substitute the values in the expression $\frac{f(x+h)-f\left(x\right)}{h}$and simplify.

$$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h}=\frac{(-3x^2-6xh-3h^2+5x+5h+2)-(-3x^2+5x+2)}{h} \\ =\frac{-6xh-3h^2+5h}{h} \\ =-6x-3h+5 \end{gathered}$$