Differentiating natural exponentials
A natural exponential remains in the derivative. If the exponent contains a coefficient of x, that coefficient is also a factor in the derivative
\[\frac{d}{dx}e^{kx}=k\,e^{kx}\]
- Keep the natural exponential The derivative of e to the power of x is the same exponential \(\displaystyle \frac{d}{dx}e^{x}=e^{x}\)
- Multiply by the inner coefficient For an exponent kx, multiply by k and keep e to the power of kx \(\displaystyle \frac{d}{dx}e^{kx}=k\,e^{kx}\)
- Differentiate each term A sum or difference of exponential terms is differentiated term by term \(\displaystyle \frac{d}{dx}\left(ae^{kx}+be^{mx}\right)=ak\,e^{kx}+bm\,e^{mx}\)
- Rewrite combined forms first Use index laws or expansion to rewrite the exponential quotients and squares used here as sums of terms before differentiating \(\displaystyle \frac{e^{3x}}{e^{x}}=e^{2x},\quad\left(2e^{x}+3\right)^{2}=4e^{2x}+12e^{x}+9\)
Worked example
Differentiate the function and find the value of its derivative at x = 0.
\[H(x)=-4e^{-2x}+3e^{x}\]
- \(\displaystyle \frac{d}{dx}\left(-4e^{-2x}\right)=(-4)(-2)e^{-2x}=8e^{-2x}\)Multiply the coefficient -4 by the inner coefficient -2.
- \(\displaystyle \frac{d}{dx}\left(3e^{x}\right)=3e^{x}\)The coefficient of x in the exponent is 1 for the second term
- \(\displaystyle H'(x)=8e^{-2x}+3e^{x}\)Combine the derivatives of the two terms
- \(\displaystyle H'(0)=8e^{0}+3e^{0}=8+3=11\)At x = 0 both exponential factors equal 1.
Common mistakes
Missing the inner coefficient
✕\(\displaystyle \frac{d}{dx}\left(3e^{2x}\right)=3e^{2x}\)✓\(\displaystyle \frac{d}{dx}\left(3e^{2x}\right)=6e^{2x}\)
The coefficient of x in the exponent is an inner derivative factor and must multiply the exponentialDropping the exponential factor
✕\(\displaystyle \frac{d}{dx}e^{3x}=3\)✓\(\displaystyle \frac{d}{dx}e^{3x}=3e^{3x}\)
The exponential is its own derivative, and the coefficient 3 from the exponent is multiplied by itChanging the square middle sign
✕\(\displaystyle \left(2e^{x}+3\right)^{2}=4e^{2x}-12e^{x}+9\)✓\(\displaystyle \left(2e^{x}+3\right)^{2}=4e^{2x}+12e^{x}+9\)
The middle term is twice the product of the two terms, so its sign is positive when both terms are positiveDifficulty levels
introDifferentiate one term of the form a e to the power of kx and evaluate at zero
standardDifferentiate a sum of two exponential terms and evaluate its exact gradient at zero
Questions about this topic
Why is the coefficient of x in the exponent needed?+
It is the derivative of the inner exponent, so the chain rule multiplies the exponential by that coefficient
Why does e to the power of x remain in the derivative?+
The natural exponential is equal to its own derivative, so differentiation changes its coefficient but keeps the exponential factor
Why rewrite a quotient before differentiating?+
When the denominator is a single exponential term, index laws simplify each numerator term into an exponential. Differentiate the resulting terms
Is my complete written rule automatically marked?+
No. Written rules and explanations are ungraded
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 6: Applications of differentiation
In this chapter
Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter