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## Implicit Differentiation

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.

# Level 2

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 $$a + b$$.

$\sin ( x + y ) = \log ( x + y )$

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 $$\frac { dy }{ dx } =$$

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.

Consider the function $$f(x) = \dfrac{x}{1+\dfrac{x}{1+\dfrac{x}{1+\ddots}}}$$. Determine the value of $$f'(0)$$.

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