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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.
(x + 2) (x + 3)
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Solution
\[{\text{ Product rule } (1}^{st} \text{ method }):\]
\[\text{ Let } u = x + 2; v = x + 3\]
\[\text{ Then }, u' = 1; v' = 1\]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( x + 2 \right)\left( x + 3 \right) \right] = \left( x + 2 \right)1 + \left( x + 3 \right)1\]
\[ = x + 2 + x + 3\]
\[ = 2x + 5\]
\[ 2^{nd} \text{ method }:\]
\[\frac{d}{dx}\left[ \left( x + 2 \right)\left( x + 3 \right) \right] = \frac{d}{dx}\left( x^2 + 5x + 6 \right)\]
\[ = 2x + 5\]
\[\text{ Using both the methods, we get the same answer }.\]
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