Practice Maths

Differentiating Composite Exponential Functions

Key Terms

The chain rule: if y = f(g(x)), then dy/dx = f′(g(x)) × g′(x).
For exponential composites: d/dx(ef(x)) = f′(x) × ef(x).
The rule says: “keep the exponential the same, multiply by the derivative of the exponent.”
This applies to any differentiable function f(x) in the exponent — including quadratics, cubics, and more.
Function y =Derivative dy/dx =
ef(x)f′(x) ef(x)
ex²2x ex²
e3x²−x(6x − 1)e3x²−x
esin xcos(x) esin x
Aef(x)Af′(x) ef(x)
Hot Tip The exponential part ef(x) never changes — you only multiply by the derivative of the exponent. Think: “ef(x) stays, f′(x) comes in front.”

Worked Example 1 — Quadratic exponent

Question: Differentiate y = ex²+3x.

Identify: outer function = eu, inner function u = x² + 3x.

Step 1: Find u′ = du/dx = 2x + 3.

Step 2: Apply chain rule: dy/dx = u′ × eu = (2x + 3)ex²+3x.

Worked Example 2 — Finding stationary points

Question: Find the x-coordinates of the stationary points of y = 2xe−x.

Note: this requires the product rule — revisit after studying that topic.

Using product rule: Let u = 2x, v = e−x. Then u′ = 2, v′ = −e−x.

dy/dx = u′v + uv′ = 2e−x + 2x(−e−x) = 2e−x − 2xe−x = 2e−x(1 − x)

Stationary point: Set dy/dx = 0. Since 2e−x > 0 always, we need 1 − x = 0, so x = 1.

The Chain Rule for Exponentials

When an exponential function has a function in its exponent rather than just x, we need the chain rule. Recall: if y = f(g(x)), we write the outer function as f(u) and the inner function as u = g(x). Then dy/dx = f′(u) × g′(x) = f′(g(x)) × g′(x).

For y = ef(x): the outer function is eu, which differentiates to eu (unchanged). The inner function is f(x), which differentiates to f′(x). So:

d/dx(ef(x)) = ef(x) × f′(x)

The exponential is preserved — it appears unchanged in the answer. The only “new” ingredient is the factor f′(x) that multiplies it.

Step-by-Step Technique

A reliable method for differentiating composite exponentials:

  1. Identify the exponent as a function f(x).
  2. Differentiate the exponent to get f′(x).
  3. Write the answer: f′(x) × ef(x).

Example: y = e4x³−2x

  1. Exponent: f(x) = 4x³ − 2x
  2. Derivative: f′(x) = 12x² − 2
  3. Answer: dy/dx = (12x² − 2)e4x³−2x

With a coefficient: y = 5ex² ⇒ dy/dx = 5 × 2x × ex² = 10xex².

Finding Stationary Points of Composite Exponentials

Stationary points occur where dy/dx = 0. For a function like y = ef(x), the derivative is f′(x)ef(x). Since ef(x) > 0 always, the product equals zero only when f′(x) = 0. So:

f′(x)ef(x) = 0 ⇒ f′(x) = 0 (since ef(x) ≠ 0)

Example: y = ex²−4x. dy/dx = (2x − 4)ex²−4x = 0 ⇒ 2x − 4 = 0 ⇒ x = 2.

At x = 2: y = e4−8 = e−4. So the stationary point is (2, e−4).

Applications and Modelling

Composite exponentials model phenomena such as:

  • Gaussian/Normal curves: y = e−x²/2 — the bell curve in statistics.
  • Logistic growth: modifications involve e raised to polynomial exponents.
  • Physics: particle wave functions, heat diffusion.

In exam context, you may be asked to find where a rate of change (derivative) equals zero or a given value, which requires setting f′(x)ef(x) = k and solving for x using logarithms.

Example: Find x when dy/dx = 4 for y = e2x².

dy/dx = 4xe2x² = 4 ⇒ xe2x² = 1

This transcendental equation has no simple closed form — CAS would be used in an exam.

Common Errors and Exam Technique

  • Forgetting the chain rule: writing d/dx(ex²) = ex² is wrong — it should be 2xex².
  • Confusing the exponent in the answer: the original exponent stays unchanged. Do NOT differentiate the exponent in the e part of the answer.
  • Incorrectly factorising: if y = ex² + e3x, the derivative is 2xex² + 3e3x. There is no further simplification; do not try to factor esomething out unless both terms share an identical exponential part.
Exam Tip: When finding stationary points of expressions involving e, always factorise first. If dy/dx = (x² − 4)ex, then dy/dx = 0 ⇒ x² − 4 = 0 (since ex ≠ 0) ⇒ x = ±2. Never divide both sides by esomething — instead set the non-exponential factor to zero.
Exam Tip: In a “show that” question, you may need to rearrange your derivative to match a given form. For example, if asked to show dy/dx = (2x − 1)ex²−x, expand carefully and check each step. Examiners award marks for correct working even if the final step has an error.

Mastery Practice

See Answers ➔

  1. Differentiate each composite exponential. Fluency

    1. (a) y = ex²
    2. (b) y = e3x²−1
    3. (c) y = e−x²
    4. (d) y = 4ex²+2x
    5. (e) y = ex³
    6. (f) y = e2x³−x

  2. Differentiate and simplify. Fluency

    1. (a) y = 3ex²−4x
    2. (b) y = −e2x²
    3. (c) y = e(x+1)²
    4. (d) y = 2e5−x²

  3. Evaluate the derivative at the given point. Fluency

    1. (a) y = ex²−1, find dy/dx at x = 1.
    2. (b) y = e2x²+x, find dy/dx at x = 0.
    3. (c) y = 3e−x², find the gradient at x = 1. Give exact value.

  4. Find and classify stationary points. Fluency

    1. (a) Find the stationary point of y = ex²−2x.
    2. (b) Find the stationary point of y = e−x²+4x−3.
    3. (c) Explain why y = ex²+1 has a stationary point but y = ex does not.

  5. Tangent and normal to a composite exponential curve. Understanding

    Consider y = ex²−3.
    1. (a) Find dy/dx.
    2. (b) Show that the curve passes through the point (√3, 1) and find the gradient of the tangent at this point.
    3. (c) Write the equation of the tangent at (√3, 1).
    4. (d) Find the gradient of the normal at (√3, 1).

  6. Bell curve differentiation. Understanding

    The standard normal PDF (bell curve) uses the function f(x) = e−x²/2.
    1. (a) Find f′(x).
    2. (b) Find the stationary point of f(x) and classify it (maximum or minimum).
    3. (c) By considering f′(x) for x > 0 and x < 0, describe the shape of the bell curve.
    4. (d) Find the x-values where f′(x) = −e−1/2. (Hint: solve from part (a).)

  7. Instantaneous rate of change in context. Understanding

    A heated object cools according to T(t) = 20 + 60e−0.1t² °C at time t minutes.
    1. (a) Find T(0) and explain its physical meaning.
    2. (b) Find T′(t).
    3. (c) Find the rate of temperature change at t = 1 and t = 3 (to 2 d.p.).
    4. (d) As t increases, what happens to T′(t)? Explain in terms of cooling rate.

  8. Gradient matching. Understanding

    The curves y = eax² and y = e3x have the same gradient at x = 1.
    1. (a) Find dy/dx for each curve in terms of a (where applicable).
    2. (b) At x = 1, set the gradients equal and solve for a.
    3. (c) Verify your answer by substituting back into both derivative expressions.

  9. Quadratic exponent application. Problem Solving

    A particle moves along a line such that its position (in metres) at time t seconds is x(t) = et²−4t for t ≥ 0.
    1. (a) Find the initial position x(0).
    2. (b) Find the velocity v(t) = x′(t).
    3. (c) Find when the particle is momentarily at rest (v = 0).
    4. (d) Find the minimum position of the particle and the time at which it occurs.
    5. (e) Determine whether the particle is moving left or right for t < 2 and t > 2.

  10. Show-that and interpretation. Problem Solving

    Let f(x) = eg(x) where g(x) is differentiable.
    1. (a) Show that f′(x) = g′(x) f(x).
    2. (b) Given g(x) = x² − 3x, find all x where f′(x) = 0 and all x where f′(x) < 0.
    3. (c) Show that f′′(x) = [g′′(x) + (g′(x))²] f(x). (Hint: differentiate f′(x) = g′(x)f(x) using the product rule.)
    4. (d) Using g(x) = x² − 3x, find f′′(x) and evaluate it at the stationary point to classify it.