The derivative from first principles
First principles forms a difference quotient for a small change h and then takes its limit. Expand and subtract before cancelling h, because substituting zero for h in the original quotient is undefined
\[f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]
- Start with the limit definition The derivative is the limit of the difference quotient as h approaches zero \(\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)
- Expand the shifted function Replace x by x + h and expand all powers before subtracting f(x) \(\displaystyle \left(x+h\right)^{2}=x^{2}+2xh+h^{2}\)
- Subtract and cancel h For nonzero h, subtract the original function, factor out h and cancel the common factor before taking the limit \(\displaystyle \frac{h\left(2x+h\right)}{h}=2x+h,\quad h\ne0\)
- Take the limit last After cancellation, let h approach zero and then substitute any requested x-value \(\displaystyle \lim_{h\to0}\left(2x+h\right)=2x\)
Worked example
Use first principles to find g'(1) for the function below
\[g(x)=x^{3}-2x+1\]
- \(\displaystyle g(x+h)=\left(x+h\right)^{3}-2\left(x+h\right)+1=x^{3}+3x^{2}h+3xh^{2}+h^{3}-2x-2h+1\)Replace x by x + h and expand the cubic
- \(\displaystyle g(x+h)-g(x)=3x^{2}h+3xh^{2}+h^{3}-2h=h\left(3x^{2}+3xh+h^{2}-2\right)\)Subtract g(x), cancel matching terms and factor out h.
- \(\displaystyle \frac{g(x+h)-g(x)}{h}=3x^{2}+3xh+h^{2}-2\quad\Rightarrow\quad g'(x)=3x^{2}-2\)Cancel h and let h approach zero in the simplified quotient
- \(\displaystyle g'(1)=3\times1^{2}-2=1\)Substitute the target value of one into the derivative rule
Common mistakes
Leaving h in the answer
✕\(\displaystyle g'(x)=3x^{2}+3xh+h^{2}-2\)✓\(\displaystyle g'(x)=3x^{2}-2\)
After the common factor of h is cancelled, the limit makes every remaining h-term zeroChanging the subtraction sign
✕\(\displaystyle g(x+h)-g(x)=3x^{2}h+3xh^{2}+h^{3}+2h\)✓\(\displaystyle g(x+h)-g(x)=3x^{2}h+3xh^{2}+h^{3}-2h\)
Subtracting the term -2x from g(x) leaves the shifted term -2h in the differenceMoving the binomial powers
✕\(\displaystyle \left(x+h\right)^{3}=x^{3}+3x^{2}h+3xh+h^{3}\)✓\(\displaystyle \left(x+h\right)^{3}=x^{3}+3x^{2}h+3xh^{2}+h^{3}\)
The middle binomial term has one x factor and two h factorsDifficulty levels
introUse a quadratic and a small integer target to form the quotient and gradient
standardUse a cubic with fractional coefficients and a rational target input
challengeObtain the derivative rule and evaluate it at a negative rational input
examSelect a quotient or derivative form and complete an exact gradient check
Questions about this topic
Why can h not be set to zero at the start?+
The difference quotient has h in the denominator, so substituting zero for h before simplifying gives division by zero
Why is the original function subtracted from the shifted function?+
The difference quotient measures the change in the output divided by the change in the input. Subtract the original output from the shifted output
Why does the first-principles result match the power rule?+
The limit definition proves the local rate of change. For polynomials, expanding and simplifying the quotient produces the same term-by-term power-rule derivative
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