NDA Maths · Application of Derivatives

Tangents, Rates of Change & Approximations

The derivative read geometrically (slope of the tangent), dynamically (a rate of change), and as a tool for estimating small changes via differentials.

All Application of Derivatives notes NDA Maths strategy

Why this matters

These are the most direct uses of f′(x): the slope of a tangent or normal, how fast one quantity changes with another, and a quick linear estimate of a small change. They are reliably easy marks once you read the derivative the right way.

Concept 1 of 2

Tangent and normal to a curve

Intuition

The slope of the tangent at a point is just

f'(x_0)

; the normal is perpendicular, so its slope is

-1/f'(x_0)

. With a point and a slope, the line equations follow immediately.

Definition

At $(x_0,y_0)$ on $y=f(x)$ : tangent slope $m=f'(x_0)$ , tangent $y-y_0=m(x-x_0)$ ; normal slope $-1/m$ , $y-y_0=-\tfrac1m(x-x_0)$ . The tangent makes angle $\theta=\tan^{-1}m$ with the x-axis. A tangent is horizontal where $f'=0$ , vertical where $f'$ is undefined; parallel tangents share the same $m$ .

Worked example

Find the slope of the tangent to

y=x^3-2x

x=1

$\dfrac{dy}{dx}=3x^2-2$ .
At $x=1$ : $3-2=1$ .

Answer:Slope

=1

Practice this conceptself-check · 4 quick reps

Try it yourself

The tangent to

x^2=y

(1,1)

makes angle

\theta

with the x-axis. Find

\tan\theta

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Tangent slope at $x_0$ ?
2.
Normal slope if tangent slope is $m$ ?
3.
Tangent is horizontal where?
4.
Angle of tangent with x-axis?

From the bank · past-year question

Example 1Application of DerivativesMODERATE

The tangent to the curve

x^{2}=y

(1,1)

makes an angle

\theta

with the positive direction of

x

-axis. Which one of the following is correct?

[Q100 · Sep · 2021]

Drill 2 more on tangent and normal to a curve

Concept 2 of 2

Rates of change and small-change approximation

Intuition

A derivative is a rate:

\dfrac{dy}{dt}

tells how fast

y

changes in time, and related quantities chain together via

\dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}

. For a small input change, the derivative gives a fast linear estimate of the output change:

\Delta y\approx f'(x)\,\Delta x

Definition

Related rates: differentiate the relation w.r.t. time and substitute known rates (e.g. radius growing → area's rate $\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}$ ).
Approximation (differentials): $\Delta y\approx \dfrac{dy}{dx}\,\Delta x$ ; use it to estimate $f(x+\Delta x)\approx f(x)+f'(x)\Delta x$ .

Worked example

The radius of a circle grows at

3

cm/s. How fast is the area changing when

r=5

cm?

$A=\pi r^2\Rightarrow\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}$ .
$=2\pi(5)(3)=30\pi$ cm $^2$ /s.

Answer:

30\pi

cm²/s.

Practice this conceptself-check · 4 quick reps

Try it yourself

For

y=3x^2+2

, estimate the change in

y

x

goes from

10

10.1

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Small-change formula?
2.
$\dfrac{dA}{dt}$ for $A=\pi r^2$ ?
3.
$y=x^2$ , $x:2\to2.01$ : $\Delta y\approx$ ?
4.
A derivative w.r.t. time is a?

From the bank · past-year question

Example 2Application of DerivativesEASY

The radius of a circle is increasing at the rate of

0\cdot7

cm/sec. What is the rate of increase of its circumference?

[Q72 · Apr · 2020]

Drill 1 more on rates of change and small-change approximation

Mastery check — 3 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Application of DerivativesEASY

The curve

y=-x^3+3x^2+2x-27

has the maximum slope at

[Q82 · Apr · 2021]

Example 2Application of DerivativesEASY

Let

y=3x^2+2

. If

x

changes from 10 to 10·1, then what is the total change in

y

[Q80 · Apr · 2020]

Example 3Application of DerivativesEASY

Where does the tangent to the curve

y=e^{x}

at the point

(0,1)

meet the

x

-axis?

[Q90 · Sep · 2021]

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