[tex]f(x) = \sqrt[3]{x} + 5 \\ [/tex]
[tex]f'(x) = \frac{d}{dx} ( \sqrt[3]{x} + 5) \\ [/tex]
[tex]f'(x) = \frac{d}{dx} (x {}^{ \frac{1}{3} } + 5) \\ [/tex]
[tex]f'(x) = \frac{d}{dx} (x {}^{ \frac{1}{3} } ) + \frac{d}{dx} (5) \\ [/tex]
[tex]f'(x) = \frac{1}{3} x {}^{ \frac{1}{3} - 1 } + 0 \\ [/tex]
[tex]f'(x) = \frac{1}{3} x {}^{ - \frac{2}{3} } + \frac{d}{dx} + 0 \\ [/tex]
[tex]f'(x) = \frac{1}{3} \: . \: \frac{1}{x {}^{ \frac{2}{3} } } \\ [/tex]
[tex]f'(x) = \frac{1}{3x {}^{ \frac{2}{3} } } \\ [/tex]
[tex]\boxed{\boxed{\boxed{f'(x) = \frac{1}{3 \sqrt[3]{x {}^{2} } } }}} \\ [/tex]
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[tex]f(x) = \sqrt[3]{x} + 5 \\ [/tex]
[tex]f'(x) = \frac{d}{dx} ( \sqrt[3]{x} + 5) \\ [/tex]
[tex]f'(x) = \frac{d}{dx} (x {}^{ \frac{1}{3} } + 5) \\ [/tex]
[tex]f'(x) = \frac{d}{dx} (x {}^{ \frac{1}{3} } ) + \frac{d}{dx} (5) \\ [/tex]
[tex]f'(x) = \frac{1}{3} x {}^{ \frac{1}{3} - 1 } + 0 \\ [/tex]
[tex]f'(x) = \frac{1}{3} x {}^{ - \frac{2}{3} } + \frac{d}{dx} + 0 \\ [/tex]
[tex]f'(x) = \frac{1}{3} \: . \: \frac{1}{x {}^{ \frac{2}{3} } } \\ [/tex]
[tex]f'(x) = \frac{1}{3x {}^{ \frac{2}{3} } } \\ [/tex]
[tex]\boxed{\boxed{\boxed{f'(x) = \frac{1}{3 \sqrt[3]{x {}^{2} } } }}} \\ [/tex]