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Derivative of $$$2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
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Find $$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right) - \frac{d}{dx} \left(2^{\frac{2}{3}} x\right) + \frac{d}{dx} \left(\sqrt[3]{2}\right)\right)}$$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 $$$c = 2^{\frac{2}{3}}$$$ and $$$f{\left(x \right)} = x$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(2^{\frac{2}{3}} x\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right) = - {\color{red}\left(2^{\frac{2}{3}} \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right)$$The derivative of a constant is $$$0$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sqrt[3]{2}\right)\right)} - 2^{\frac{2}{3}} \frac{d}{dx} \left(x\right) + \frac{d}{dx} \left(2 x^{2}\right) = {\color{red}\left(0\right)} - 2^{\frac{2}{3}} \frac{d}{dx} \left(x\right) + \frac{d}{dx} \left(2 x^{2}\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 2^{\frac{2}{3}} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(2 x^{2}\right) = - 2^{\frac{2}{3}} {\color{red}\left(1\right)} + \frac{d}{dx} \left(2 x^{2}\right)$$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 $$$c = 2$$$ and $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right)\right)} - 2^{\frac{2}{3}} = {\color{red}\left(2 \frac{d}{dx} \left(x^{2}\right)\right)} - 2^{\frac{2}{3}}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} - 2^{\frac{2}{3}} = 2 {\color{red}\left(2 x\right)} - 2^{\frac{2}{3}}$$Thus, $$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right) = 4 x - 2^{\frac{2}{3}}$$$.
Answer
$$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right) = 4 x - 2^{\frac{2}{3}}$$$A