Q. 55 - Calculus

Question

Use the definition of the derivative to find the derivatives described in Exercises 55-58.

Find $\frac{d}{dx}\left(2x^3\right),\frac{d^2}{d{x}^2}\left(2x^3\right),\frac{d^3}{d{x}^3}\left(2x^3\right)$

Step-by-Step Solution

Answer

The derivatives are $6x^2,12x,12$ respectively

Step 1. Given information

Given function $y=2x^3$

Step 2: Use the definition of derivative and calculate

Calculating, we get

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{2(x+h{)}^3-2x^3}{h} \\ \underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}==\underset{h\to 0}{\lim }\frac{2\left(x^3+h^3+3x^{2}h+3x{h}^2\right)-2x^3}{h} \\ ==\underset{h\to 0}{\lim }\frac{2x^3+2h^3+6x^{2}h+6x{h}^2-2x^3}{h} \\ =\underset{h\to 0}{\lim }\frac{2h^3+6x^{2}h+6x{h}^2}{h} \\ =\underset{h\to 0}{\lim }\frac{h\left(2h^2+6x^2+6xh\right)}{h} \\ =\underset{h\to 0}{\lim }\left(2h^2+6x^2+6xh\right) \\ =2(0{)}^2+6x^2+6x(0) \\ =6x^2 \\ \frac{d}{dx}\left(2x^3\right)=6x^2 \end{gathered}$$

Step 3: Calculating the other derivatives

Calculating, we get

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{g(x+h)-g\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{6(x+h{)}^2-6x^2}{h} \\ \underset{h\to 0}{\lim }\frac{g(x+h)-g\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{6\left(x^2+h^2+2xh\right)-6x^2}{h} \\ =\underset{h\to 0}{\lim }\frac{6x^2+6h^2+12xh-6x^2}{h} \\ =\underset{h\to 0}{\lim }\frac{6h^2+12xh}{h} \\ =\underset{h\to 0}{\lim }(6h+12x) \\ =12x \\ \frac{d^2}{d{x}^2}\left(2x^3\right)=12x \end{gathered}$$

Step 4: Calculating the other derivatives

Calculating, we get

$$\begin{gathered} \underset{h\to 0}{\lim }\frac{m(x+h)-m\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{12(x+h)-12x}{h} \\ \underset{h\to 0}{\lim }\frac{m(x+h)-m\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{12x+12h-12x}{h} \\ =\underset{h\to 0}{\lim }\frac{12h}{h} \\ =\underset{h\to 0}{\lim }12 \\ =12 \\ \frac{d^3}{d{x}^3}\left(2x^3\right)=12 \end{gathered}$$