Question 28.7.2: If y=1/√a²-x², find d y/dx....
If y=\frac{1}{\sqrt{a^{2}-x^{2}}} , find \frac{d y}{d x} .
Step-by-Step
Put \left(a^{2}-x^{2}\right)=t , so that y=\frac{1}{\sqrt{t}}=t^{-1 / 2} and t=\left(a^{2}-x^{2}\right) .
\therefore \quad \frac{d y}{d t}=-\frac{1}{2} t^{-3 / 2} and \frac{d t}{d x}=-2 x .
So, \quad \frac{d y}{d x}=\left(\frac{d y}{d t} \times \frac{d t}{d x}\right) \\ \\=\left(-\frac{1}{2} t^{-3 / 2}\right)(-2 x)=x t^{-3 / 2}=x\left(a^{2}-x^{2}\right)^{-3 / 2} .
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