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Find $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right)$$$

The calculator will find $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right)$$$, with steps shown.

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Find $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)$$$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(3 x^{2}\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 3$$$:

$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - \frac{d}{dx} \left(3 x^{2}\right) = {\color{red}\left(3 x^{2}\right)} - \frac{d}{dx} \left(3 x^{2}\right)$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 3$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$3 x^{2} - {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = 3 x^{2} - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$3 x^{2} - 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 x^{2} - 3 {\color{red}\left(2 x\right)}$$

Simplify:

$$3 x^{2} - 6 x = 3 x \left(x - 2\right)$$

Thus, $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right)$$$.

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

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 3$$$ and $$$f{\left(x \right)} = x \left(x - 2\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(3 x \left(x - 2\right)\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x \left(x - 2\right)\right)\right)}$$

Apply the product rule $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ with $$$f{\left(x \right)} = x$$$ and $$$g{\left(x \right)} = x - 2$$$:

$$3 {\color{red}\left(\frac{d}{dx} \left(x \left(x - 2\right)\right)\right)} = 3 {\color{red}\left(\frac{d}{dx} \left(x\right) \left(x - 2\right) + x \frac{d}{dx} \left(x - 2\right)\right)}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$3 x {\color{red}\left(\frac{d}{dx} \left(x - 2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$

The derivative of a constant is $$$0$$$:

$$3 x \left(- {\color{red}\left(\frac{d}{dx} \left(2\right)\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$3 x {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 3 \left(x - 2\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x {\color{red}\left(1\right)} + 3 \left(x - 2\right) {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(3 x \left(x - 2\right)\right) = 6 x - 6$$$.

Next, $$$\frac{d^{3}}{dx^{3}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(6 x - 6\right)$$$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dx} \left(6 x - 6\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(6 x\right) - \frac{d}{dx} \left(6\right)\right)}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 6$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}\left(\frac{d}{dx} \left(6 x\right)\right)} - \frac{d}{dx} \left(6\right) = {\color{red}\left(6 \frac{d}{dx} \left(x\right)\right)} - \frac{d}{dx} \left(6\right)$$

The derivative of a constant is $$$0$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(6\right)\right)} + 6 \frac{d}{dx} \left(x\right) = - {\color{red}\left(0\right)} + 6 \frac{d}{dx} \left(x\right)$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$6 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 6 {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(6 x - 6\right) = 6$$$.

Next, $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(6\right)$$$

The derivative of a constant is $$$0$$$:

$${\color{red}\left(\frac{d}{dx} \left(6\right)\right)} = {\color{red}\left(0\right)}$$

Thus, $$$\frac{d}{dx} \left(6\right) = 0$$$.

Therefore, $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = 0$$$.

Answer

$$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = 0$$$A

Related Notes: Definition of Derivative , Derivatives of Elementary Functions , Table of Derivatives , Related Rates , Studying Derivative Graphically , Optimization Problems , Applications to Economics , Constant Multiple Rule , Sum and Difference Rules , Product Rule , Quotient Rule , Chain Rule , Derivative of Inverse Function