If x=4z^2+5,y=6z^2+7z+3find d^2y dx^2 . - Vidyadip

Question

If $\text{x}=4\text{z}^2+5,\text{y}=6\text{z}^2+7\text{z}+3$find $\frac{\text{d}^2\text{y}}{\text{dx}^2}.$

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Answer

Here,
$\text{x}=4\text{z}^2+5,\text{y}=6\text{z}^2+7\text{z}+3$
Differentiating w.r.t.x, we get
$\frac{\text{dx}}{\text{dz}}=8\text{z}\ \text{and}\ \frac{\text{dy}}{\text{dz}}=12\text{z}+7$
$\therefore\frac{\text{dy}}{\text{dx}}=\frac{12\text{z}+7}{8\text{z}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{12\times8\text{z}-8(12\text{z}+7)}{64\text{z}^2}\times\frac{\text{dz}}{\text{dx}}$
$=\frac{96\text{z}-96\text{z}-56}{512\text{z}^3}=\frac{-7}{64\text{z}^3}$

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