Exercise
given the following function:
y=\frac{x}{\sqrt{9-x^2}}-f(\frac{x}{3})
The following holds:
f'(x)=\frac{1}{\sqrt{1-x^2}}
Find the derivative of y.
Final Answer
Solution
y=\frac{x}{\sqrt{9-x^2}}-f(\frac{x}{3})
Using Derivative formulas and the quotient rule and chain rule in Derivative Rules, we get the derivative:
y'=\frac{\sqrt{9-x^2}-x\cdot\frac{1}{2\sqrt{9-x^2}}\cdot (-2x)}{9-x^2}-\frac{1}{\sqrt{1-{(\frac{x}{3})}^2}}\cdot \frac{1}{3}=
הערה: השתמשנו בנגזרת של f הנתונה בשאלה וכפלנו בנגזרת הפנימית לפי כלל ההרכבה.
Note: We used the derivative of f given in the question and multiplied the internal derivative by the chain rule in Derivative Rules.
We simplify the derivative:
y'=\frac{\sqrt{9-x^2}+\frac{2x^2}{2\sqrt{9-x^2}}}{9-x^2}-\frac{1}{\sqrt{1-\frac{x^2}{9}}}\cdot \frac{1}{3}=
=\frac{\frac{2(9-x^2)+2x^2}{2\sqrt{9-x^2}}}{9-x^2}-\frac{1}{3\sqrt{\frac{9-x^2}{9}}}=
=\frac{9-x^2+x^2}{(9-x^2)\sqrt{9-x^2}}-\frac{1}{\sqrt{9-x^2}}=
=\frac{9-(9-x^2)}{\sqrt{{(9-x^2)}^3}}=
=\frac{x^2}{\sqrt{{(9-x^2)}^3}}
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