Applications of Derivatives
What the derivative is for — measuring how fast things change, finding tangents, locating the highest and lowest points of a curve, and turning slope into a tool for solving real problems
- The derivative as a rate of change, and how to link two changing quantities with related rates.
- How to write the equations of the tangent and normal to a curve at a point.
- Using the sign of \(f'\) to decide where a function is increasing or decreasing.
- Locating maxima and minima with the first- and second-derivative tests.
- Rolle's theorem and the Mean Value Theorem, and what they guarantee.
- Concavity and points of inflection, approximations with differentials, and L'Hôpital's rule.
Rates of Change
The derivative \(\dfrac{dy}{dx}\) is the rate at which \(y\) changes per unit change in \(x\). When several quantities vary together in time, each carries its own rate, and the chain rule ties them into a single equation — these are related-rate problems, like a ladder sliding down a wall or a balloon being inflated.
Differentiate the relation between the quantities with respect to time. Each variable contributes its own time-derivative, and known rates pin down the unknown one. The instantaneous rate \(\left.\tfrac{dy}{dx}\right|_{x=a}\) is evaluated at the moment of interest, after differentiating — never before.
Tangents & Normals
At a point \((x_1,y_1)\) on a curve, the derivative gives the slope of the tangent. The normal is the line through the same point perpendicular to the tangent, so its slope is the negative reciprocal.
The tangent uses the slope \(m=f'(x_1)\); the normal uses \(-1/m\). A horizontal tangent occurs where \(f'(x_1)=0\), and a vertical tangent where \(f'(x_1)\) is infinite — there the roles of tangent and normal simply swap.
Increasing & Decreasing Functions
Where the tangent slopes upward the function is rising; where it slopes downward the function is falling. So the sign of the first derivative reads off the direction of the curve, and the points where it changes sign are exactly the turning points.
On an interval where \(f'>0\) throughout, the function is strictly increasing; where \(f'<0\) throughout, strictly decreasing. To find the intervals, solve \(f'(x)=0\), mark those points on a number line, and test the sign of \(f'\) in each gap.
Maxima & Minima
A local maximum is a peak and a local minimum is a valley; at either, the tangent is horizontal, so \(f'(c)=0\). Such a point is called a critical point. To classify it, either watch how \(f'\) changes sign, or check the sign of \(f''\).
First-derivative test: if \(f'\) changes \(+\to-\) at \(c\) it is a maximum, \(-\to+\) a minimum. Second-derivative test: the sign of \(f''(c)\) decides it directly. If \(f''(c)=0\) the test is inconclusive — fall back to the first-derivative test.
On a closed interval \([a,b]\), the absolute maximum and minimum occur either at a critical point or at an endpoint. So evaluate \(f\) at every critical point and at \(a\) and \(b\), then simply pick the largest and smallest values.
Rolle's & the Mean Value Theorem
Two existence theorems connect the average behaviour of a function to its instantaneous slope. Rolle's theorem is the special case where the endpoints sit at the same height; the Mean Value Theorem (MVT) is the tilted general version.
If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there is at least one interior point \(c\) where the tangent matches the average rate of change. Rolle is the level case; the MVT says the instantaneous slope equals the secant slope somewhere — the geometric heart of many calculus proofs.
Concavity & Points of Inflection
The second derivative measures how the slope itself is turning. Where \(f''>0\) the curve bends upward like a cup (concave up); where \(f''<0\) it bends downward like a cap (concave down). The boundary between the two is a point of inflection.
A point of inflection is where the concavity reverses — so \(f''=0\) and the sign of \(f''\) actually changes there. Note that \(f''=0\) alone is not enough: at \(y=x^4\) the second derivative vanishes at the origin, yet the curve stays concave up.
Approximations & Differentials
Near a point, a smooth curve looks almost like its tangent line. That linear approximation lets you estimate a nearby value, or gauge how a small error in a measurement propagates into a computed quantity.
The differential \(dy=f'(x)\,dx\) is the change predicted along the tangent, and it approximates the true change \(\Delta y\) for small \(\Delta x\). It is the everyday tool for estimates such as \(\sqrt{4.1}\), and for error analysis: a relative error \(\tfrac{dx}{x}\) in a measurement feeds forward as \(\tfrac{dy}{y}\) in the result.
L'Hôpital's Rule
When a limit collapses to the indeterminate form \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), derivatives come to the rescue: differentiate top and bottom separately and try the limit again.
The rule applies only to the forms \(\tfrac{0}{0}\) and \(\tfrac{\infty}{\infty}\) — confirm the form before using it. Other forms such as \(0\cdot\infty\), \(\infty-\infty\) or \(1^{\infty}\) must first be rewritten as a quotient. The rule may be applied repeatedly if the new limit is still indeterminate.
Putting It to Work
Problem. A circle's radius grows at \(3\ \text{cm/s}\). How fast is the area growing when \(r=5\ \text{cm}\)?
Solution. From \(A=\pi r^2\), differentiate with respect to time:
Problem. Find the tangent and normal to \(y=x^2\) at \((1,1)\).
Solution. Here \(f'(x)=2x\), so the tangent slope is \(m=2\) and the normal slope is \(-\tfrac12\):
Problem. Where is \(f(x)=x^3-3x\) increasing or decreasing?
Solution. \(f'(x)=3x^2-3=3(x-1)(x+1)\). Testing the sign across \(x=\pm1\):
Problem. Classify the extrema of \(f(x)=x^3-3x\).
Solution. Critical points \(x=\pm1\); with \(f''(x)=6x\):
Problem. Verify Rolle's theorem for \(f(x)=x^2-4x+3\) on \([1,3]\).
Solution. It is a polynomial (continuous and differentiable) and \(f(1)=f(3)=0\). Solve \(f'(c)=0\):
Problem. Estimate \(\sqrt{4.1}\) using a differential.
Solution. Take \(f(x)=\sqrt{x}\) at \(x=4\), where \(f(4)=2\) and \(f'(4)=\tfrac14\):
Chapter Summary
Differentiate the relation in time; \(\tfrac{dy}{dt}=\tfrac{dy}{dx}\tfrac{dx}{dt}\) links related rates.
Slope \(m=f'(x_1)\) for the tangent, \(-1/m\) for the normal.
\(f'>0\) increasing, \(f'<0\) decreasing; turning points at \(f'=0\).
Critical \(f'(c)=0\); classify with \(f''(c)\) or the sign of \(f'\).
Rolle \(f'(c)=0\); MVT \(f'(c)=\tfrac{f(b)-f(a)}{b-a}\).
\(f''\) sets concavity; \(dy=f'(x)\,dx\) approximates; L'Hôpital for \(\tfrac00,\tfrac\infty\infty\).
Problems
Name the application first — rate, tangent, monotonicity, extremum, mean value, approximation — then apply the matching tool. Difficulty rises down the list.
- Find the slope of the tangent to \(y=x^2-3x+2\) at \(x=2\).
- Find the equation of the normal to \(y=x^3\) at \((1,1)\).
- The side of a square grows at \(2\ \text{cm/s}\). How fast is its area growing when the side is \(10\ \text{cm}\)?
- Find the intervals on which \(f(x)=2x^3-3x^2-12x+5\) is increasing.
- Find the local maximum and minimum values of \(f(x)=x^3-6x^2+9x+1\).
- Find the absolute maximum of \(f(x)=x^3-3x\) on \([0,2]\).
- A spherical balloon's radius grows at \(0.5\ \text{cm/s}\). Find \(\dfrac{dV}{dt}\) when \(r=7\ \text{cm}\).
- Verify Rolle's theorem for \(f(x)=\sin x\) on \([0,\pi]\).
- Find the value of \(c\) in the Mean Value Theorem for \(f(x)=x^2\) on \([1,3]\).
- Use differentials to approximate \(\sqrt{25.3}\).
- Find the point of inflection of \(f(x)=x^3-3x^2+1\).
- Find two positive numbers whose sum is \(20\) and whose product is a maximum.
- Evaluate \(\displaystyle\lim_{x\to0}\frac{\sin x}{x}\) using L'Hôpital's rule.