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Differentiate natural exponential functions and solve simple derivative conditions

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Which first rule should be used for the displayed exponential?

\[f(x)=3e^{2x}\]
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

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

\[\frac{d}{dx}e^{kx}=k\,e^{kx}\]
  1. Keep the natural exponential The derivative of e to the power of x is the same exponential \(\displaystyle \frac{d}{dx}e^{x}=e^{x}\)
  2. Multiply by the inner coefficient For an exponent kx, multiply by k and keep e to the power of kx \(\displaystyle \frac{d}{dx}e^{kx}=k\,e^{kx}\)
  3. Differentiate each term A sum or difference of exponential terms is differentiated term by term \(\displaystyle \frac{d}{dx}\left(ae^{kx}+be^{mx}\right)=ak\,e^{kx}+bm\,e^{mx}\)
  4. Rewrite combined forms first Use index laws or expansion to rewrite the exponential quotients and squares used here 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 zero

\[H(x)=-4e^{-2x}+3e^{x}\]
  1. \(\displaystyle \frac{d}{dx}\left(-4e^{-2x}\right)=(-4)(-2)e^{-2x}=8e^{-2x}\)Multiply the coefficient -4 by the inner coefficient -2
  2. \(\displaystyle \frac{d}{dx}\left(3e^{x}\right)=3e^{x}\)The coefficient of x in the exponent is 1 for the second term
  3. \(\displaystyle H'(x)=8e^{-2x}+3e^{x}\)Combine the derivatives of the two terms
  4. \(\displaystyle H'(0)=8e^{0}+3e^{0}=8+3=11\)At zero, 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 exponential
Dropping 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 it
Changing 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 positive

Difficulty 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
challengeRewrite a quotient or expanded square before differentiating
examSolve an exact derivative condition and select a tangent or normal equation

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 gives the same exponential multiplied by k

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

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