Calculus
# Implicit Differentiation

Consider the function

\[ f(x) = \sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}. \]

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

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

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.

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

Consider the function \(\displaystyle f(x) = \frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}. \)

Determine the value of \( f'(0).\)

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