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Q)

Find the differential $dy$ and evaluate $dy$ for the given values of $x$ and $dx$\[\] $y$$=1-x^{2},x$$=5,dx$$=\large\frac{1}{2}$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

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A)
Toolbox:
  • Let $y=f(x)$ be a differentiable function then the quantities $dx$ and $dy$ are called differentials.The differential $dx$ is an independent variable.
  • The differential $dy$ is then defined by $dy=f'(x)dx(dx \approx \Delta x)$
  • Also $f(x+\Delta x)-f(x)=\Delta y =dy $ from which $f(x+\Delta x) $can be evaluated
Step 1:
Given $y=1-x^2$
$dy=-2xdx$
Step 2:
When $x=5,dx=\large\frac{1}{2}$
$dy=-2(5) \large\frac{1}{2}$$=-5$
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