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Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.

Consider the function

\( f(x) = \sqrt{x+\sqrt{x+\sqrt{x+\ldots}}} \)

If the gradient of the tangent of the function at \( x = 12 \) can be expressed as \( \frac{a}{b} \) where \( a \) and \( b \) are positive coprime integers, what is the value of \( a + b \).

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\[ \sin ( x + y ) = \log ( x + y ) \]

Find the value of \( \frac {dy}{dx}\).

Note that we didn't give you the base of the logarithm, you should not specifically assume that the logarithm is in base \(e\) or 10.

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If \({ x }^{ y }={ e }^{ x-y }\), then \(\frac { dy }{ dx } =\)

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Which of the given equations is the common tangent to the circle \({(x-3)}^2+{y}^2=9\) and the parabola \({y}^2=4x\) touching both the conics above the \(x\)-axis.

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