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Question
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
Solution
\[\frac{dy}{dx} = \frac{d}{dx}\left( \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x \right)\]
\[ = \frac{2}{3}\frac{d}{dx}\left( x^9 \right) - \frac{5}{7}\frac{d}{dx}\left( x^7 \right) + 6\frac{d}{dx}\left( x^3 \right) - \frac{d}{dx}\left( x \right)\]
\[ = \frac{2}{3}\left( 9 x^8 \right) - \frac{5}{7}\left( 7 x^6 \right) + 6\left( 3 x^2 \right) - 1\]
\[ = 6 x^8 - 5 x^6 + 18 x^2 - 1\]
\[\frac{dy}{dx} at x = 1:\]
\[6 \left( 1 \right)^8 - 5 \left( 1 \right)^6 + 18 \left( 1 \right)^2 - 1\]
\[ = 6 - 5 + 18 - 1\]
\[ = 18\]
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