The graph of a derivative
Where it exists, the derivative records the tangent slope of the original function. Stationary points become zeros of the derivative; positive and negative tangent slopes give its sign
\[f'(x)=0\text{ at the stationary x-values of }f\]
- Use stationary points as zeros The x-coordinate of a stationary point of f is an x-intercept of f' \(\displaystyle f'(a)=0\quad\text{at a stationary input }x=a\)
- Read signs from slope A positive tangent slope gives a positive derivative and a negative tangent slope gives a negative derivative. An increasing curve can still have zero derivative at an isolated stationary input \(\displaystyle \begin{aligned}\text{positive tangent slope}&\Rightarrow f'(x)>0\\\text{negative tangent slope}&\Rightarrow f'(x)<0\end{aligned}\)
- Reduce polynomial degree Differentiating a non-constant polynomial reduces its degree by one \(\displaystyle \deg(f)=3\quad\Rightarrow\quad\deg(f')=2\)
- Exclude non-smooth inputs A finite derivative requires a well-defined instantaneous slope. A corner or jump prevents a derivative; continuity alone does not guarantee one \(\displaystyle x=a\text{ is a corner}\quad\Rightarrow\quad f'(a)\text{ does not exist}\)
Worked example
Use the function features to state the key features of its derivative
\[h(x)=x^{3}-6x^{2}+9x\]
- \(\displaystyle h'(x)=3x^{2}-12x+9=3(x-1)(x-3)\)Differentiate the cubic and factorise the derivative
- \(\displaystyle h'(x)=0\quad\Rightarrow\quad x=1,3\)The stationary x-values of h are the zeros of h'
- \(\displaystyle h'(0)=9>0,\quad h'(2)=-3<0,\quad h'(4)=9>0\)The derivative sign pattern from left to right is positive, negative, positive
- \(\displaystyle \deg(h)=3\quad\Rightarrow\quad\deg(h')=2\)The derivative graph has degree one less than the cubic function
Common mistakes
Reading sign from function height
✕\(\displaystyle f(x)=x^{3}-3x+2,\quad f(-2)=0\quad\Rightarrow\quad f'(x)<0\quad\text{for }x<-1\)✓\(\displaystyle f'(x)=3x^{2}-3>0\quad\text{for }x<-1\)
The derivative sign comes from the slope of the function, not from its height above the x-axisUsing function intercepts as derivative zeros
✕\(\displaystyle f(x)=x^{3}-3x+2,\quad f(x)=0\quad\Rightarrow\quad x=-2,1\quad\text{are the zeros of }f'\)✓\(\displaystyle f'(x)=0\quad\Rightarrow\quad x=-1,1\quad\text{from the stationary inputs}\)
The derivative is zero at stationary points, whereas function intercepts are where f itself is zeroKeeping the original degree
✕\(\displaystyle \deg(f)=3\quad\Rightarrow\quad\deg(f')=3\)✓\(\displaystyle \deg(f)=3\quad\Rightarrow\quad\deg(f')=2\)
Differentiation reduces the degree of a non-constant polynomial by oneDifficulty levels
introRead a cubic with two integer turning points and determine derivative signs
standardUse a quartic with several turning points to identify derivative intercepts
challengeSelect a qualitative derivative graph using degree, zeros and sign pattern
examUse a piecewise function to determine where the derivative exists
Questions about this topic
Where does the derivative graph cross the x-axis?+
It crosses or touches the x-axis at x-values where the original function has a stationary point, because the derivative is zero there
What determines whether f' is positive?+
A positive tangent slope gives a positive derivative, regardless of the function height. An increasing function need not have a strictly positive derivative at every input: the cube function has a horizontal tangent at zero
Why can a derivative be undefined?+
At a corner or jump, there is no well-defined finite instantaneous slope. Continuity alone is not enough to guarantee a derivative; for example, the continuous absolute-value graph has a corner at zero
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 5: Differentiation
In this chapter
Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method