Derivatives represent the instantaneous rate of change of a function. This has various applications in real-world scenarios.
Steps to Solve Contextual Problems Involving Derivatives
- Identify the function: Determine the function that describes the relationship between the variables involved.
- Identify the variable you want to differentiate: This will depend on the question being asked.
- Calculate the derivative: Use the appropriate differentiation rules to find the derivative of the function.
- Interpret the result: Explain the meaning of the derivative in the context of the problem.
Implicit Differentiation
Implicit Differentiation is a technique used to find the derivative of a function that is not explicitly defined as $ y = f(x) $ . Instead, the function is defined by an equation that implicitly relates $ x $ and $ y $ , or any other variable you may see.
Examples
Example 1: Find the slope of the tangent line to the curve defined by the equation $ x^2 + y^2 = 25 $ at the point $ (3, 4) $ .
- Differentiate both sides:
$$ 2x + 2y \frac{dy}{dx} = 0 $$ 2. Solve for $ \frac{dy}{dx} $ :
$$ \frac{dy}{dx} = -\frac{x}{y} $$ 3. Substitute the point (3, 4):
$$ \frac{dy}{dx} = -\frac{3}{4} $$
Therefore, the slope of the tangent line at the point $ (3, 4) $ is $ -\frac{3}{4} $ .
Example 2: Find the rate of change of the radius of a circle with respect to its area, given that the area is increasing at a rate of $ 10 \pi $ square units per second.
- Identify the equation: The area of a circle is given by $ A = \pi r^2 $ .
- Differentiate both sides:
$$ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} $$ 3. Substitute the given information:
$$ 10 \pi = 2\pi r \frac{dr}{dt} $$ 4. Solve for $ \frac{dr}{dt} $ :
$$ \frac{dr}{dt} = \frac{5}{r} $$
Therefore, the rate of change of the radius with respect to its area is $ \frac{5}{r} $ units per second.