Differentiate trigonometric and exponential functions
Use the standard derivatives of sine, cosine, tangent, the exponential and the natural logarithm, with trigonometric inputs in radians.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In which angle unit are calculus trigonometric functions measured in 0606?
What you will learn
- Differentiate sin x, cos x and tan x
- Differentiate eˣ and ln x
- Use radians for every trigonometric differentiation
Know the standard function pairs
d(sin x)/dx = cos x; d(cos x)/dx = −sin x; d(tan x)/dx = sec²x. For exponential and logarithmic functions, d(eˣ)/dx = eˣ and d(ln x)/dx = 1/x for x > 0.
These are standard derivatives to recall and apply. In the 0606 syllabus trigonometric angles for calculus are always measured in radians; the sine derivative rule does not have this form when the input is in degrees.
Apply constants and sums term by term
For y = 4 sin x − 3 cos x + 2 tan x, differentiate each function and keep its coefficient: y′ = 4 cos x + 3 sin x + 2 sec²x.
The domain of ln x requires x > 0. The tangent derivative is defined only where cos x ≠ 0, matching the original function's domain.
Pause and explain
What is d(tan x)/dx?
Worked example
Differentiate y = 3 sin x + 2 cos x + eˣ + ln x.
Show the worked solution
- The derivative of 3 sin x is 3 cos x.
- The derivative of 2 cos x is −2 sin x; the derivative of eˣ is eˣ.
- The derivative of ln x is 1/x for x > 0.
Answer dy/dx = 3 cos x − 2 sin x + eˣ + 1/x, for x > 0.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Differentiate y = 4 sin x − 3 cos x + 2 tan x.
Give me a hint
Apply each standard trigonometric derivative in radians.
Compare my reasoning
- 4 sin x becomes 4 cos x.
- −3 cos x becomes +3 sin x.
- 2 tan x becomes 2 sec²x.
dy/dx = 4 cos x + 3 sin x + 2 sec²x.
Look for these in your work
- Sine derivative used
- Cosine sign changed correctly
- Tangent square retained
Differentiate y = 2eˣ − 5 ln x for x > 0.
Give me a hint
The exponential is unchanged; the logarithm differentiates to 1/x.
Compare my reasoning
- d(2eˣ)/dx = 2eˣ.
- d(−5 ln x)/dx = −5/x.
- Combine the terms and retain the domain.
dy/dx = 2eˣ − 5/x, for x > 0.
Look for these in your work
- Exponential derivative correct
- Logarithm derivative correct
- Domain condition preserved
A periodic model is y = 6 cos x + 3 tan x. Find dy/dx at x = π/4.
Give me a hint
Differentiate first, then use exact trigonometric values in radians.
Compare my reasoning
- dy/dx = −6 sin x + 3 sec²x.
- At π/4, sin = √2/2 and sec² = 2.
- Substitute and simplify.
dy/dx at π/4 = 6 − 3√2.
Look for these in your work
- Derivative rules applied
- Radians retained
- Exact values substituted
Common mistakes
- Forgetting the negative sign in the cosine derivative
- Using tan x derivative as sec x rather than sec²x
- Using degree mode for trigonometric calculus
- Applying 1/x for ln x where x is not positive
What can you explain now?
What is d(cos x)/dx?
Compare with the explanation
−sin x
The derivative of cosine is negative sine.
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Make it stick
What are the standard derivatives of sin, cos, tan, eˣ and ln x, and what angle unit is required?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.