Find the Third Derivative f(x)=2x^3-7x+1

$f (x) = 2 x^{3} - 7 x + 1$

Step 1

Find the first derivative.

By the Sum Rule, the derivative of $2 x^{3} - 7 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [1]$ .

$f' (x) = \frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (1)$

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (1)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 2 (3 x^{2}) + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (1)$

Multiply $3$ by $2$ .

$f' (x) = 6 x^{2} + \frac{d}{d x} (- 7 x) + \frac{d}{d x} (1)$

Evaluate $\frac{d}{d x} [- 7 x]$ .

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$f' (x) = 6 x^{2} - 7 \frac{d}{d x} x + \frac{d}{d x} (1)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 6 x^{2} - 7 \cdot 1 + \frac{d}{d x} (1)$

Multiply $- 7$ by $1$ .

$f' (x) = 6 x^{2} - 7 + \frac{d}{d x} (1)$

Differentiate using the Constant Rule.

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f' (x) = 6 x^{2} - 7 + 0$

Add $6 x^{2} - 7$ and $0$ .

$f' (x) = 6 x^{2} - 7$

Step 2

Find the second derivative.

By the Sum Rule, the derivative of $6 x^{2} - 7$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 7]$ .

$f'' (x) = \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (- 7)$

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 7)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 6 (2 x) + \frac{d}{d x} (- 7)$

Multiply $2$ by $6$ .

$f'' (x) = 12 x + \frac{d}{d x} (- 7)$

Differentiate using the Constant Rule.

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f'' (x) = 12 x + 0$

Add $12 x$ and $0$ .

$f'' (x) = 12 x$

Step 3

Find the third derivative.

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$f''' (x) = 12 \frac{d}{d x} (x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f''' (x) = 12 \cdot 1$

Multiply $12$ by $1$ .

$f''' (x) = 12$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $12$ .

$12$

Find the Third Derivative f(x)=2x^3-7x+1

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