Chapter 6: Application of Derivatives

Rate of Change, Monotonicity & Optimization

Basic Definition	$$\frac{dy}{dx}$$represents the rate of change of $$y $$ with respect to $$x$$.
Value at a specific point	$$\left. \frac{dy}{dx} \right\|_{x=x_0}$$represents rate of change at$$x = x_0$$.
Chain Rule Application	If $$y$$and$$x$$both change with time$$t$$: $$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$$
Marginal Cost (MC)	Rate of change of Total Cost $$C(x)$$w.r.t output$$x$$: $$MC = \frac{dC}{dx}$$
Marginal Revenue (MR)	Rate of change of Total Revenue $$R(x)$$w.r.t sales$$x$$: $$MR = \frac{dR}{dx}$$

Strictly Increasing	$$f'(x) > 0$$for all$$x$$ in the interval.
Increasing	$$f'(x) \ge 0$$for all$$x$$ in the interval.
Strictly Decreasing	$$f'(x) < 0$$for all$$x$$ in the interval.
Decreasing	$$f'(x) \le 0$$for all$$x$$ in the interval.
Neither Increasing nor Decreasing	If $$f'(x)$$ changes sign (positive to negative or vice versa) within the interval.

Critical Points	Points where $$f'(x) = 0$$or$$f'(x)$$ is undefined.
Second Derivative Test (Step 1)	Find $$x$$such that$$f'(x) = 0$$. Let these points be $$c_1, c_2 \dots$$
Local Maxima Condition	If $$f”(c) < 0$$, then $$x = c$$ is a point of Local Maxima.
Local Minima Condition	If $$f”(c) > 0$$, then $$x = c$$ is a point of Local Minima.
Test Failure (Inflection)	If $$f”(c) = 0$$, the test fails (use First Derivative Test). It might be a Point of Inflection.
First Derivative Test (Maxima)	If $$f'(x)$$changes sign from Positive (+) to Negative (-) as$$x$$increases through$$c$$.
First Derivative Test (Minima)	If $$f'(x)$$changes sign from Negative (-) to Positive (+) as$$x$$increases through$$c$$.