Implicit differentiation

Find ${\displaystyle \frac{dy}{dx}}$ where $y$ is defined implicitly by $y^4+x^3{y}^2-x=5$.

Equation of tangent line via implicit differentiation

Find the equation of the line tangent to the curve defined implicitly by $x^{4}y-x^3+14y=29$ at the point $(1,2)$.

Elasticity of demand using implicit differentiation

The demand equation for Hager & Little’s Better Business Calculus book series is given by

where $p$ is the wholesale unit price in dollars and $x$ is the quantity demanded in units of a thousand. Compute the elasticity of demand when $x=4$.

Related rates problem: demand vs. revenue

A company is increasing production of math-brain protein bars at a rate of 50 cases per day. All cases produced can be sold. The daily demand function is given by

where $x$ is the number of cases produced and sold, and $p$ is in dollars. Find the rate of change of the revenue with respect to the time in days when daily production is 500 cases.

Related rates problem: price vs. demand

If the quantity demanded daily of a product is related to its unit price in dollars by

how fast is the quantity demanded changing when $x=5$ and the unit price is decreasing at a rate of $3 per day?

Related rates problem: price vs. demand

The weekly demand function is given by

where $x$ is the number of thousands of units demanded weekly and $p$ is in dollars. If the price is increasing at a rate of 20 cents per week when the level of demand is 4000 units, at what rate is the demand decreasing?