Find the second derivative using implicit differentiation for the equation:

$3x^2 - 2y^2 = 6xy - 4$

Using implicit differentiation, what is the $\frac{dy}{dx}$ of the function below?

$x^3+y^3=15xy$

Find the slope of the tangent line to the curve at the given point:

${(3x^2+2y^2)}^2=9{(x+y)}^2$ at $(-1,0)$

Assume the cost, in dollars, to manufacture $x$ bicycles is given by the function $c\left(x\right)=500+50x-0.05x^2$. Determine the marginal cost when $200$ bicycles are produced, and verify it by calculating the cost of producing one additional bicycle after the first $200$.

A hot air balloon is descending vertically at a rate of $3\text{ ft/s}$. If there is a person standing $40\text{ ft}$ away from the point right below the balloon, how fast is the distance between the person and the balloon decreasing when the balloon is $30\text{ ft}$ above the ground?

Analyze if the piecewise function satisfies the Mean Value Theorem criteria over the domain of the function.

A cyclist goes from a standstill to a speed of $45\text{ km/hr}$ in 15 seconds. Calculate the minimum value of the cyclist's maximum acceleration during this period in $\text{km/hr/s}$.

Determine the linearization of the given function at the specified value of x.

$f\left(x\right)=\frac{3}{2+x}+\sqrt{4-x}-5$ at $x=0$

Find the linear approximation of the function $g(x)$ for small values of $x$ using the formula $(1 + x)^k ≈ 1 + kx$.

$g(x)=(1+x)^4$

A cyclist starts from rest and completes a $15\text{-mi}$ race in $45\text{ minutes}$. Suppose the cyclist's speed at the finish line is zero, explain why the cyclist must have been riding at exactly $20\text{ mph}$ at least twice during the race.

Evaluate the limit of the given expression. Use l'Hôpital's Rule if required.

${\lim }_{x\to 2^{-}}\frac{{\sin }^{-1}(x-1)-\frac{\pi }{2}}{x-2}$

Identify which of the two functions grows faster. Use limit methods.

$x^3\ln x$ and $x^4$

Find the value of $h^{\prime }\left(e\right)$ where $h\left(x\right)={\left(3x\right)}^{\frac{1}{3x}}$, and $e$ is the base of the natural logarithm.

Calculate the derivative of the function $y=\frac{(2x^2-3{)}^5}{(x^6+5{)}^7(4x+1{)}^6}.$

Use logarithmic differentiation to evaluate the derivative of the function $g\left(x\right)=x^{e^x}$.