Calculating Derivative – Deriving a function in another function – Exercise 6279


given the following function:


The following holds:


Find the derivative of y.

Final Answer




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}=





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