For the equation $$x*y + x^2*y - y^3 = 0$$, rank the following steps in applying implicit differentiation to solve for $$\frac{dy}{dx}$$ (from first to last): A) Differentiate each term with respect to $$x$$, applying the product rule to terms like $$x*y$$ and $$x^2*y$$. B) For terms involving $$y$$, apply the chain rule (i.e. multiply by $$\frac{dy}{dx}$$). C) Collect all terms containing $$\frac{dy}{dx}$$ on one side. D) Factor out $$\frac{dy}{dx}$$ and solve for it.

I. Given $$f(x)=x^3+2*x$$, if $$f(1)=3$$ and $$f'(1)=5$$, then for its inverse function $$g=f^{-1}$$ we have $$g'(3)=\frac{1}{5}$$. II. In general, the derivative of an inverse function is given by $$\frac{d}{dx}[f^{-1}(x)]=f'(x)$$. III. For an invertible function $$f$$, the derivative of its inverse at $$f(a)$$ is $$\frac{1}{f'(a)}$$. Which of the above statements is/are true regarding inverse function differentiation?

Differentiate $$y= \sqrt{1+3*x^2}$$ with respect to x. Which of the following represents the correct derivative $$y'$$?

All of the following statements regarding the derivative of polynomial functions are true except:

A particle’s position is given by $$s(t)=4*t^2+3*t-5$$ (in meters). Compute the acceleration (in m/s²) of the particle at t = 3 seconds.

A spherical balloon is being inflated such that its volume increases at a rate of $$10$$ cubic inches per second. Using the volume formula $$V=\frac{4}{3}\pi*r^3$$, find the rate at which the radius is increasing when $$r=4$$ inches.

By the Mean Value Theorem, for the function $$f(x)= x^2$$ on the interval [3, 4], find the value c in (3, 4) where $$f'(c)$$ equals the average rate of change.

Evaluate the limit $$\lim_{x\to 0}\frac{\sin(2*x)}{x}$$ using L'Hôpital's Rule.

Which of the following is an example of using derivatives to locate horizontal tangent lines? (A horizontal tangent occurs where the derivative is equal to zero.)