Find the Third Derivative f(x)=x^4-10x^3

$f (x) = x^{4} - 10 x^{3}$

Step 1

Find the first derivative.

Differentiate.

By the Sum Rule, the derivative of $x^{4} - 10 x^{3}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 10 x^{3}]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 10 x^{3})$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 10 x^{3})$

Evaluate $\frac{d}{d x} [- 10 x^{3}]$ .

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{3}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 x^{3} - 10 \frac{d}{d x} x^{3}$

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

$f' (x) = 4 x^{3} - 10 (3 x^{2})$

Multiply $3$ by $- 10$ .

$f' (x) = 4 x^{3} - 30 x^{2}$

Step 2

Find the second derivative.

By the Sum Rule, the derivative of $4 x^{3} - 30 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 30 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 30 x^{2})$

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 30 x^{2})$

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

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 30 x^{2})$

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 30 x^{2})$

Evaluate $\frac{d}{d x} [- 30 x^{2}]$ .

Since $- 30$ is constant with respect to $x$ , the derivative of $- 30 x^{2}$ with respect to $x$ is $- 30 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 30 \frac{d}{d x} x^{2}$

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

$f'' (x) = 12 x^{2} - 30 (2 x)$

Multiply $2$ by $- 30$ .

$f'' (x) = 12 x^{2} - 60 x$

Step 3

Find the third derivative.

By the Sum Rule, the derivative of $12 x^{2} - 60 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 60 x]$ .

$f''' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 60 x)$

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f''' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 60 x)$

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

$f''' (x) = 12 (2 x) + \frac{d}{d x} (- 60 x)$

Multiply $2$ by $12$ .

$f''' (x) = 24 x + \frac{d}{d x} (- 60 x)$

Evaluate $\frac{d}{d x} [- 60 x]$ .

Since $- 60$ is constant with respect to $x$ , the derivative of $- 60 x$ with respect to $x$ is $- 60 \frac{d}{d x} [x]$ .

$f''' (x) = 24 x - 60 \frac{d}{d x} x$

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

$f''' (x) = 24 x - 60 \cdot 1$

Multiply $- 60$ by $1$ .

$f''' (x) = 24 x - 60$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $24 x - 60$ .

$24 x - 60$

Find the Third Derivative f(x)=x^4-10x^3

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