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Question
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
Solution
Here,
\[x = 4 z^2 + 5 \text { and y } = 6 z^2 + 7z + 3\]
\[\text { Differentiating w . r . t . z, we get }\]
\[\frac{d x}{d z} = 8z and \frac{d y}{d z} = 12z + 7\]
\[ \therefore \frac{d y}{d x} = \frac{12z + 7}{8z}\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = \frac{12 \times 8z - 8\left( 12z + 7 \right)}{64 z^2} \times \frac{dz}{dx}\]
\[ = \frac{96z - 96z - 56}{512 z^3} = \frac{- 7}{64 z^3}\]
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