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MCV4U Derivatives of Exponential & Trigonometric Functions
Grade 12 Unit 4 — practice questions with full solutions

Differentiating exponential, logarithmic and trigonometric functions, and combining them with the product and chain rules.

Free MCV4U derivatives of exponential & trigonometric functions practice for Grade 12 students in Ontario. These 6 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MCV4U unit test, catching up on a lesson, or reviewing before the final exam.
MCV4U · Grade 12 Derivatives and their Applications 5 core 1 challenge
1Core
Differentiate: y = e3x2.
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  1. For y = eu, the chain rule gives y′ = eu × u′.
  2. Here u = 3x2, so u′ = 6x.
  3. Therefore y′ = 6xe3x2.
Final answery′ = 6xe3x2
2Core
Differentiate: y = ln(5x2 + 1).
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  1. For y = ln(u), the chain rule gives y′ = u′ ÷ u.
  2. Here u = 5x2 + 1, so u′ = 10x.
  3. Therefore y′ = 10x ÷ (5x2 + 1).
Final answery′ = 10x ÷ (5x2 + 1)
3Core
Differentiate each: (a) y = sin(4x) and (b) y = cos2x.
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  1. (a) The derivative of sin(u) is cos(u) × u′, and here u′ = 4, so y′ = 4cos(4x).
  2. (b) Write cos2x as (cos x)2 and use the chain rule.
  3. The outer power gives 2(cos x), and the derivative of cos x is −sin x.
  4. So y′ = −2 cos x sin x, which the double-angle identity writes as −sin(2x).
Final answer(a) 4cos(4x)   (b) −2 cos x sin x = −sin(2x)
4Core
Differentiate y = x2ex and write the answer in factored form.
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  1. This is a product, so use y′ = u′v + uv′.
  2. Let u = x2 so u′ = 2x; let v = ex so v′ = ex.
  3. y′ = 2xex + x2ex.
  4. Factor out the common xex: y′ = xex(2 + x).
Final answery′ = xex(x + 2)
5Core
A colony is modelled by P(t) = 500e0.08t, where t is in hours. Find the rate of growth at t = 10 hours, to one decimal place.
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  1. Differentiate: P′(t) = 500(0.08)e0.08t = 40e0.08t.
  2. At t = 10 the exponent is 0.8, so P′(10) = 40e0.8.
  3. e0.8 ≈ 2.2255, so P′(10) ≈ 40(2.2255) ≈ 89.0.
  4. The colony is growing at about 89 organisms per hour at that moment.
Final answer≈ 89.0 per hour
6Challenge
Differentiate y = e2xsin(3x) and write the answer in factored form.
Show the full solution
  1. Product rule with u = e2x and v = sin(3x).
  2. u′ = 2e2x by the chain rule.
  3. v′ = 3cos(3x), also by the chain rule.
  4. y′ = 2e2xsin(3x) + 3e2xcos(3x).
  5. Factor out e2x: y′ = e2x[2sin(3x) + 3cos(3x)].
Final answery′ = e2x[2sin(3x) + 3cos(3x)]

MCV4U Derivatives of Exponential & Trigonometric Functions — common questions

Short answers to the things students ask most about this unit.

What is the derivative of e raised to a function?

It is the same exponential multiplied by the derivative of the exponent. For example the derivative of e^(3x²) is 6x·e^(3x²).

Why does the derivative of cos²x become −sin(2x)?

The chain rule gives −2·cos x·sin x, and the double-angle identity sin(2x) = 2·sin x·cos x rewrites that as −sin(2x). Both forms are correct.

More MCV4U practice — grade 12 calculus and vectors

MCV4U Rates of Change & LimitsMCV4U DerivativesMCV4U Applications of Derivatives & Curve SketchingMCV4U Geometric VectorsAll MCV4U units
← Unit 3: Applications of Derivatives & Curve SketchingUnit 5: Geometric Vectors →
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