f(x)=3x^2 find f(a+h)-f(a) / h
You asked:
Evaluate the function \(f\left( x \right) = 3 {x}^{2}\), \(f\left( a + h \right) - \frac{f\left( a \right)}{h}\).
MathBot Answer:
$$\begin{aligned}f\left( a + h \right) - \frac{f\left( a \right)}{h}&=\frac{3 a^{2} h - 3 a^{2} + 6 h^{2} a + 3 h^{3}}{h}\end{aligned}$$
$$\begin{aligned}f\left( a + h \right) - \frac{f\left( a \right)}{h}&=3 \left(a + h\right)^{2} - \frac{3 a^{2}}{h}\\f\left( a + h \right) - \frac{f\left( a \right)}{h}&=\frac{\left(-3\right) a^{2}}{h} + 3 \left(a + h\right)^{2}\\f\left( a + h \right) - \frac{f\left( a \right)}{h}&=\frac{\left(-3\right) a^{2}}{h} + 3 \left(a^{2} + 2 a h + h^{2}\right)\\f\left( a + h \right) - \frac{f\left( a \right)}{h}&=\frac{3 a^{2} h - 3 a^{2} + 6 h^{2} a + 3 h^{3}}{h}\\\end{aligned}$$