Calculus

Implicit Differentiation

Implicit Differentiation: Level 2 Challenges

         

Consider the function

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

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

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

Find the value of dydx \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 ee or 10.

If xy=exy{ x }^{ y }={ e }^{ x-y }, then dydx=_______________.\frac { dy }{ dx } = \text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}.

Which of the given equations is the common tangent to the circle (x3)2+y2=9{(x-3)}^2+{y}^2=9 and the parabola y2=4x{y}^2=4x touching both the conics above the xx-axis?

Consider the function f(x)=x1+x1+x1+.\displaystyle f(x) = \frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}.

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

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