Derivatives and optimisation

Idea: The slope behind the question

Where a curve peaks, how steep it is, where a tangent meets the axis: all use the slope at a point.

• Timed tests mostly use polynomials
• Product rule: $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
• A few rules and one careful habit cover nearly every question

Formula: Power rule

$(x^n{)}^{\prime }=n{x}^{n-1}$

Differentiate term by term; a constant factor stays, a constant term disappears.

Example: Slope of a cubic

Find the slope of $f(x)=2x^3-5x^2+4x-7$ at $x=2$.

1. $f^{\prime }(x)=6x^2-10x+4$
2. At $x=2$: $6\times 4-10\times 2+4=8$

Formula: Chain rule

$(g(h(x)){)}^{\prime }=g^{\prime }(h(x))\times h^{\prime }(x)$

Differentiate the outside, keep the inside, then multiply by the derivative of the inside.

Common mistake: Forgetting the inside

For $(3x-1{)}^4$ writing $4(3x-1{)}^3$ and stopping, without the factor $3$.

Instead Multiply by the inside's derivative: at $x=1$ the slope is $12\times 8=96$, not $32$.

Formula: The tangent line

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

At x = 0 it crosses at f(a) − a f′(a), a favourite quick question.

Quick check: Where the tangent crosses

For $f(x)=2x^3-5x^2+4x-7$, where does the tangent at $x=2$ cross the y axis?

Idea: Extremes on an interval

On $[a,b]$ the largest and smallest values sit at a critical point inside or at an endpoint.

• Solve $f^{\prime }(x)=0$, keep the roots inside $[a,b]$
• Evaluate $f$ there and at both endpoints
• Pick the largest and smallest on the list

Example: The largest value on [−2, 6]

Find the largest value of $f(x)=x^3-3x^2-9x+5$ on $[-2,6]$.

1. $f^{\prime }(x)=3(x-3)(x+1)$: critical points $-1$ and $3$
2. $f(-1)=10$, $f(3)=-22$, $f(-2)=3$
3. Endpoint $f(6)=216-108-54+5=59$ is the largest

Common mistake: Reporting the local maximum

The local maximum at $x=-1$ gives $10$, yet the largest value is $59$ at an endpoint.

Instead Always put the endpoints on the list, and drop any root of $f^{\prime }$ outside the interval.

Idea: Say it in the room

Say the derivative out loud, term by term, then substitute.

• Tangent: state the point and the slope first
• Optimum: list the candidates, endpoints included
• Word problem: one variable and its allowed range