Applications of Derivatives

The One Thing

Derivatives measure instantaneous change and provide the principal method for analysing tangent and normal lines, increasing and decreasing behaviour, and maxima and minima. In optimization, solving $f^{\prime }(x)=0$ identifies candidates, but all relevant critical points and endpoints must be examined before a conclusion is reached.

Definitions and Results

• Derivative: The derivative of $y=f(x)$ at $x$ is the instantaneous rate of change of $y$ with respect to $x$, written as $\frac{dy}{dx}$ or $f^{\prime }(x)$:

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h},$$ when the limit exists.

• Geometrical meaning of derivative: At a point on a curve, $f^{\prime }(x)$ represents the slope or gradient of the tangent.

• Tangent: A straight line that touches a curve at a point and has the same slope as the curve there.

• Normal: A line perpendicular to the tangent at the point of contact.

• Slope of tangent: For $y=f(x)$ at $x=a$,

$$m_t=f^{\prime }(a).$$

• Slope of normal: If $f^{\prime }(a)\ne 0$, the normal has slope

$$m_n=-\frac{1}{f^{\prime }(a)}.$$

• Equation of tangent: At $(a,f(a))$,

$$y-f(a)=f^{\prime }(a)(x-a).$$

• Equation of normal: Provided $f^{\prime }(a)\ne 0$,

$$y-f(a)=-\left[\frac{1}{f^{\prime }(a)}\right](x-a).$$

• Horizontal tangent: If $f^{\prime }(a)=0$, the tangent is horizontal. The normal is vertical and has equation

$$x=a.$$

• Vertical tangent: If the derivative is undefined or infinite, the usual slope form is not applicable. The tangent may be represented by

$$x=a.$$

• Rate of change: $\frac{dy}{dx}$ gives the instantaneous rate of change of $y$ with respect to $x$. Its units are the units of the dependent quantity divided by the units of the independent quantity.

• Chain rule: If $y=f(u)$ and $u=g(x)$, then

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

• Related rates: Variables connected by an equation are differentiated with respect to time $t$, after which known values are substituted.

• Increasing function: A function is increasing on an interval when

$$f^{\prime }(x)>0$$ throughout that interval.

• Decreasing function: A function is decreasing on an interval when

$$f^{\prime }(x)<0$$ throughout that interval.

• Critical point: A point in the domain where $f^{\prime }(x)=0$ or where $f^{\prime }(x)$ does not exist, provided the function is defined there.

• Stationary point: A point where

$$f^{\prime }(x)=0.$$ It may be a maximum, a minimum, or neither.

• Local maximum: $f$ has a local maximum at $x=a$ if its value there is greater than nearby function values.

• Local minimum: $f$ has a local minimum at $x=a$ if its value there is less than nearby function values.

• First derivative test: A change in $f^{\prime }(x)$ from positive to negative indicates a local maximum. A change from negative to positive indicates a local minimum. No sign change generally indicates neither.

• Second derivative test: If

$$f^{\prime }(a)=0,$$ then: $$f^{\prime \prime }(a)<0\Rightarrow \text{local maximum},$$ $$f^{\prime \prime }(a)>0\Rightarrow \text{local minimum}.$$ If $f^{\prime \prime }(a)=0$, the test gives no definite conclusion.

• Absolute maximum or minimum: The greatest or least value of a function over its entire specified domain or interval.

• Optimization: The process of finding the greatest or least possible value of a quantity by forming a function and examining its critical points and endpoints.

• Absolute extrema on $[a,b]$: Evaluate the function at every critical point in $(a,b)$ and at the endpoints $x=a$ and $x=b$, then compare the values.

• Approximation using differentials: For a small change $dx$,

$$dy=f^{\prime }(x)dx,$$ so $$f(x+dx)\approx f(x)+f^{\prime }(x)dx.$$

• Functions of two variables: In basic optimization settings, partial derivatives may be used. Single-variable maxima–minima problems are primarily handled using ordinary derivatives.

• Applications: Derivatives can be used to find instantaneous velocity from displacement, marginal cost from cost, maximum area or volume, minimum distance, and greatest or least profit.

Worked Methods

Finding a derivative from the definition

• Begin with

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.$$

• Substitute the function $f$ into the expression.
• Simplify the quotient.
• Evaluate the limit as $h\to 0$, provided the limit exists.

Finding the tangent and normal to $y=f(x)$

• Identify the point on the curve, written as $(a,f(a))$.
• Differentiate $y=f(x)$ to obtain $f^{\prime }(x)$.
• Evaluate the derivative at $x=a$:

$$m_t=f^{\prime }(a).$$

• Use the tangent equation:

$$y-f(a)=f^{\prime }(a)(x-a).$$

• If $f^{\prime }(a)\ne 0$, calculate the normal slope:

$$m_n=-\frac{1}{f^{\prime }(a)}.$$

• Use the normal equation:

$$y-f(a)=-\left[\frac{1}{f^{\prime }(a)}\right](x-a).$$

• If $f^{\prime }(a)=0$, state that the tangent is horizontal and the normal is the vertical line $x=a$.

Using the chain rule

• Express the function in terms of an intermediate variable:

$$y=f(u),u=g(x).$$

• Find $\frac{dy}{du}$.
• Find $\frac{du}{dx}$.
• Multiply:

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

Solving a related-rates problem

• Write an equation connecting the relevant variables.
• Differentiate the equation with respect to time $t$.
• Substitute the known values, including the relevant rates and measurements.
• Solve for the required rate.
• State the answer with appropriate units.

Determining intervals of increase and decrease

• Find $f^{\prime }(x)$.
• Solve $f^{\prime }(x)=0$ and identify points where $f^{\prime }(x)$ does not exist, provided the function is defined there.
• Use these values to divide the domain into intervals.
• Determine the sign of $f^{\prime }(x)$ on each interval.
• State that the function is increasing where $f^{\prime }(x)>0$ and decreasing where $f^{\prime }(x)<0$.

Classifying local extrema using the first derivative test

• Find the critical points.
• Construct a sign chart for $f^{\prime }(x)$ on either side of each critical point.
• A change from $+$ to $-$ gives a local maximum.
• A change from $-$ to $+$ gives a local minimum.
• If there is no sign change, the stationary point is generally neither a maximum nor a minimum.

Classifying local extrema using the second derivative test

• Find the critical point $x=a$ by solving $f^{\prime }(a)=0$.
• Calculate $f^{\prime \prime }(a)$.
• If $f^{\prime \prime }(a)<0$, conclude that $f$ has a local maximum at $a$.
• If $f^{\prime \prime }(a)>0$, conclude that $f$ has a local minimum at $a$.
• If $f^{\prime \prime }(a)=0$, use the first derivative test or higher-order analysis instead of drawing an automatic conclusion.

Finding absolute extrema on a closed interval

• Find $f^{\prime }(x)$.
• Identify all critical points in the open interval $(a,b)$.
• Evaluate the function at each critical point.
• Evaluate the function at both endpoints, $x=a$ and $x=b$.
• Compare all resulting values.
• The greatest value is the absolute maximum and the least value is the absolute minimum on $[a,b]$.

Solving an optimization problem

• Define the quantity to be maximized or minimized.
• Form a function in one variable, using the relevant conditions and restrictions.
• State the permissible domain.
• Differentiate the function.
• Find critical points by solving $f^{\prime }(x)=0$ and checking where $f^{\prime }(x)$ does not exist.
• Classify the candidates using the first derivative test or second derivative test.
• Check endpoints and any other domain restrictions.
• Compare the candidate values and state the greatest or least value, including appropriate units.

Approximating using differentials

• Identify the function $y=f(x)$.
• Determine the initial value of $x$ and the small change $dx$.
• Calculate $f^{\prime }(x)$.
• Use

$$dy=f^{\prime }(x)dx.$$

• Approximate the new value using

$$f(x+dx)\approx f(x)+f^{\prime }(x)dx.$$

Where It Goes Wrong

• Using $-1/f^{\prime }(a)$ for the normal slope when $f^{\prime }(a)=0$; in that case the tangent is horizontal and the normal is vertical, with equation $x=a$.
• Applying the usual slope form to a vertical tangent, where the derivative is undefined or infinite and the tangent may instead be represented by $x=a$.
• Treating every solution of $f^{\prime }(x)=0$ as a maximum or minimum; a stationary point may be neither.
• Forgetting that critical points also include points where $f^{\prime }(x)$ does not exist, provided the function is defined there.
• Using the second derivative test when $f^{\prime \prime }(a)=0$; this gives no definite conclusion, so the first derivative test or higher-order analysis is required.
• Solving $f^{\prime }(x)=0$ in an optimization problem but failing to check endpoints, domain restrictions, units, or the final comparison of candidate values.

What Gets Asked

• Find a derivative from

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}.$$

• Interpret a derivative as an instantaneous rate of change or as the slope of a tangent.
• Find the equations of the tangent and normal to a curve at a specified point.
• Handle horizontal and vertical tangents and identify the corresponding normal.
• Apply the chain rule to composite functions.
• Solve related-rates problems by differentiating a connecting equation with respect to time.
• Determine intervals on which a function is increasing or decreasing using the sign of $f^{\prime }(x)$.
• Locate critical and stationary points.
• Classify local maxima and minima using the first derivative test or second derivative test.
• Explain why $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)=0$ do not by themselves establish an extremum.
• Find absolute maxima and minima on a closed interval by checking critical points and endpoints.
• Solve optimization problems involving maximum area or volume, minimum distance, or greatest or least profit.
• Use differentials to approximate a changed function value.
• Interpret derivative units in physical and practical applications, including instantaneous velocity from displacement and marginal cost from cost.
• Recognize that functions of two variables may require partial derivatives in basic optimization settings.