Derivatives as rates: velocity, acceleration, and chain rule with time

Key idea

In applications, the derivative often means an instantaneous rate of change. If position is \(s(t)\), then: \[ v(t)=s'(t)\quad \text{(velocity)}, \qquad a(t)=v'(t)=s''(t)\quad \text{(acceleration)}. \] When variables depend on time, the chain rule connects rates: \[ \frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}. \]

Worked example

Try it

Summary

• Velocity is \(s'(t)\); acceleration is \(s''(t)\).
• For time-dependent variables, use \(\dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}\).

Related rates: set up an equation, then differentiate with respect to time

Key idea

Related rates problems follow a consistent process:

• Step 1: Draw a picture and write a relationship among variables (often geometry).
• Step 2: Differentiate both sides with respect to time \(t\).
• Step 3: Substitute the values at the instant and solve for the unknown rate.

Worked example

Try it

Summary

• Write a relationship first (geometry), then differentiate with respect to time.
• Plug in the instant values last and solve for the unknown rate.

Optimization: maximize or minimize with derivatives

Key idea

Optimization problems follow a reliable checklist:

• Define variables and write the quantity you want to optimize (objective function).
• Use a constraint to rewrite the objective using one variable.
• Differentiate and solve \(A'(x)=0\) (or \(R'(p)=0\), etc.) to find critical points.
• Confirm max/min (use a test or the problem context).

Worked example

Try it

Summary

• Write the objective, use the constraint, then differentiate and solve for critical points.
• Use context or a test to confirm you found a maximum or minimum.

Critical points, increasing/decreasing, and the first derivative test

Key idea

Derivative tests turn graphs into algebra:

• Critical points: solve \(f'(x)=0\) (and include where \(f'(x)\) is undefined).
• Increasing/decreasing: if \(f'(x)>0\), \(f\) increases; if \(f'(x)<0\), \(f\) decreases.
• First derivative test: if \(f'\) changes from \(+\) to \(-\), you have a local maximum; from \(-\) to \(+\), a local minimum.

Worked example

Try it

Summary

• Critical points come from \(f'(x)=0\) (and where \(f'\) is undefined).
• Sign of \(f'(x)\) tells where \(f\) increases/decreases; sign changes locate local extrema.

Linear approximation: the tangent line as a fast estimator

Key idea

Near a point \(x=a\), a differentiable function behaves almost like its tangent line: \[ f(x)\approx f(a)+f'(a)(x-a). \] This is powerful for quick mental estimates and for understanding error sensitivity.

Worked example

Try it

Summary

• Linear approximation uses the tangent line: \(f(x)\approx f(a)+f'(a)(x-a)\).
• Pick \(a\) where \(f(a)\) and \(f'(a)\) are easy to compute.

Average vs instantaneous change: the Mean Value Theorem idea

Key idea

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then the Mean Value Theorem says there exists at least one number \(c\in(a,b)\) such that \[ f'(c)=\frac{f(b)-f(a)}{b-a}. \] This is the formal bridge from an average slope to a tangent slope.

Worked example

Try it

Summary

• MVT guarantees at least one point where the tangent slope equals the average slope on an interval.
• Chain rule rates often appear alongside applications: \(\dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}\).

Why applications of derivatives matter

Where applications of derivatives show up

• Physics: velocity, acceleration, and rates of change in time.
• Economics: maximize revenue, minimize cost, find optimal pricing.
• Engineering: related rates, sensitivity, and design optimization under constraints.
• Data & modeling: derivatives measure how outputs respond to inputs (local behavior).

Worked example: volume changing in a cylinder

Try it

Final recap

• Motion: \(v(t)=s'(t)\), \(a(t)=s''(t)\).
• Chain rule rates: \(\dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}\).
• Related rates: write an equation, differentiate with respect to \(t\), then substitute values.
• Optimization: objective + constraint \(\rightarrow\) one variable \(\rightarrow\) derivative \(\rightarrow\) check.
• Derivative tests: critical points and sign of \(f'(x)\) reveal where a function increases/decreases.
• Linear approximation: \(f(x)\approx f(a)+f'(a)(x-a)\) near \(a\).