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Question
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3x2 + 2)2
Solution
\[ {\text{ Product rule } (1}^{st} \text{ method }):\]
\[\text{ Let } u = 3 x^2 + 2; v = 3 x^2 + 2\]
\[\text{ Then }, u' = 6x; v' = 6x\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( 3 x^2 + 2 \right)\left( 3 x^2 + 2 \right) \right] = \left( 3 x^2 + 2 \right)\left( 6x \right) + \left( 3 x^2 + 2 \right)\left( 6x \right)\]
\[ = 18 x^3 + 12x + 18 x^3 + 12x\]
\[ = 36 x^3 + 24x\]
\[ 2^{nd} \text{ method }:\]
\[\frac{d}{dx}\left[ \left( 3 x^2 + 2 \right)^2 \right] = \frac{d}{dx}\left( 9 x^4 + 12 x^2 + 4 \right)\]
\[ = 36 x^3 + 24x\]
\[\text{ Using both the methods, we get the same answer }.\]
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