Find the Second Derivative y=(8x^3)/3-7x

$y = \frac{8 x^{3}}{3} - 7 x$

Step 1

Find the first derivative.

By the Sum Rule, the derivative of $\frac{8 x^{3}}{3} - 7 x$ with respect to $x$ is $\frac{d}{d x} [\frac{8 x^{3}}{3}] + \frac{d}{d x} [- 7 x]$ .

$f' (x) = \frac{d}{d x} (\frac{8 x^{3}}{3}) + \frac{d}{d x} (- 7 x)$

Evaluate $\frac{d}{d x} [\frac{8 x^{3}}{3}]$ .

Since $\frac{8}{3}$ is constant with respect to $x$ , the derivative of $\frac{8 x^{3}}{3}$ with respect to $x$ is $\frac{8}{3} \frac{d}{d x} [x^{3}]$ .

$f' (x) = \frac{8}{3} \cdot \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 7 x)$

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

$f' (x) = \frac{8}{3} \cdot (3 x^{2}) + \frac{d}{d x} (- 7 x)$

Combine $3$ and $\frac{8}{3}$ .

$f' (x) = \frac{3 \cdot 8}{3} x^{2} + \frac{d}{d x} (- 7 x)$

Multiply $3$ by $8$ .

$f' (x) = \frac{24}{3} x^{2} + \frac{d}{d x} (- 7 x)$

Combine $\frac{24}{3}$ and $x^{2}$ .

$f' (x) = \frac{24 x^{2}}{3} + \frac{d}{d x} (- 7 x)$

Cancel the common factor of $24$ and $3$ .

Factor $3$ out of $24 x^{2}$ .

$f' (x) = \frac{3 (8 x^{2})}{3} + \frac{d}{d x} (- 7 x)$

Cancel the common factors.

Factor $3$ out of $3$ .

$f' (x) = \frac{3 (8 x^{2})}{3 (1)} + \frac{d}{d x} (- 7 x)$

Cancel the common factor.

$f' (x) = \frac{3 (8 x^{2})}{3 \cdot 1} + \frac{d}{d x} (- 7 x)$

Rewrite the expression.

$f' (x) = \frac{8 x^{2}}{1} + \frac{d}{d x} (- 7 x)$

Divide $8 x^{2}$ by $1$ .

$f' (x) = 8 x^{2} + \frac{d}{d x} (- 7 x)$

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

$f' (x) = 8 x^{2} - 7 \frac{d}{d x} x$

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

$f' (x) = 8 x^{2} - 7 \cdot 1$

Multiply $- 7$ by $1$ .

$f' (x) = 8 x^{2} - 7$

Step 2

Find the second derivative.

By the Sum Rule, the derivative of $8 x^{2} - 7$ with respect to $x$ is $\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 7]$ .

$f'' (x) = \frac{d}{d x} (8 x^{2}) + \frac{d}{d x} (- 7)$

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 8 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 7)$

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

$f'' (x) = 8 (2 x) + \frac{d}{d x} (- 7)$

Multiply $2$ by $8$ .

$f'' (x) = 16 x + \frac{d}{d x} (- 7)$

Differentiate using the Constant Rule.

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

$f'' (x) = 16 x + 0$

Add $16 x$ and $0$ .

$f'' (x) = 16 x$

Find the Second Derivative y=(8x^3)/3-7x

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