Once the basic derivative rules are secure, Additional Mathematics extends differentiation to trigonometric, exponential and logarithmic functions. The key is to preserve the function’s structure while applying the correct derivative pattern and, when necessary, the chain rule.
These functions are especially important because they connect algebra, graph behaviour and calculus. Their derivatives often retain the same function family, which makes them powerful for modelling changing quantities.
1. Trigonometric derivatives use radians
The standard derivative results for sine and cosine are based on angles measured in radians:
- d/dx(sin x)=cos x
- d/dx(cos x)=−sin x
- d/dx(tan x)=sec²x, where defined
Using degrees would introduce an extra conversion factor. In calculus, radian measure is the natural angle system for these clean derivative formulas.
2. Worked trig derivative
Differentiate y=4sin x−3cos x.
dy/dx=4cos x+3sin x.
The negative sign in the cosine derivative changes the sign of the second term.
3. Trig plus chain rule
Differentiate y=sin(3x).
Outer derivative gives cos(3x). Inner derivative of 3x is 3. Therefore dy/dx=3cos(3x).
Similarly, d/dx[cos(5x−2)]=−5sin(5x−2).
4. Products involving trig functions
Differentiate y=x²sin x.
Use the product rule:
dy/dx=2xsin x+x²cos x.
The power rule and trig derivative operate on different factors inside the same product-rule structure.
5. Exponential functions
The natural exponential function has the distinctive property
d/dx(e^x)=e^x.
Its rate of change equals its current value. This is one reason e appears naturally in growth and decay models.
6. Exponential plus chain rule
Differentiate y=e^(4x−1).
Outer derivative remains e^(4x−1). Multiply by inner derivative 4:
dy/dx=4e^(4x−1).
7. General exponential base
For a positive constant a, a≠1,
d/dx(a^x)=a^x ln a.
Thus d/dx(2^x)=2^xln2. The factor ln a distinguishes general exponential bases from e.
8. Natural logarithm
For x>0,
d/dx(ln x)=1/x.
The positive domain is part of the original real logarithm function and must be preserved in interpretation.
9. Logarithm plus chain rule
Differentiate y=ln(3x+5).
Derivative of ln u is u′/u, so
dy/dx=3/(3x+5), where 3x+5>0 for the original real-valued function.
10. Logarithms can simplify products before differentiation
Sometimes a product or quotient of positive expressions can be rewritten using logarithm laws before differentiating. At this level, choose this only when it genuinely simplifies the structure; direct product and quotient rules may be clearer.
For example, if y=ln(x²+1), direct chain rule gives 2x/(x²+1). No extra logarithmic manipulation is needed.
11. Mixed-function worked example
Differentiate y=e^x cos x.
Use product rule:
dy/dx=e^xcos x−e^xsin x=e^x(cos x−sin x).
Factoring e^x helps reveal structure and can simplify later stationary-point work.
12. Another mixed example
Differentiate y=ln x·sin x for x>0.
Product rule gives
dy/dx=(sin x)/x+ln x cos x.
13. Derivative signs and graph behaviour
For y=e^x, the derivative e^x is always positive, so the function is increasing for every real x. For y=ln x, derivative 1/x is positive for x>0, so the logarithm is increasing throughout its domain.
These derivative facts align with the known graph shapes and provide an independent check.
14. Exact structure before approximation
If a derivative contains e^2, ln3 or exact trigonometric values, keep them exact until the question requests a decimal. Exact forms preserve relationships and reduce rounding drift.
15. Common mistakes
- Writing derivative of cos x as sin x instead of −sin x.
- Forgetting the chain-rule multiplier in sin(kx), e^(kx) or ln(kx+c).
- Differentiating a^x as xa^(x−1), which confuses exponential and power forms.
- Forgetting logarithm domain restrictions.
- Using degree-mode intuition inside calculus formulas that assume radians.
16. Independent practice
- Differentiate 5sin x+2cos x.
- Differentiate cos(4x).
- Differentiate e^(2x+3).
- Differentiate 3^x.
- Differentiate ln(5x−1).
- Differentiate xe^x.
- Differentiate x²cos x.
17. Answers
- 5cos x−2sin x.
- −4sin(4x).
- 2e^(2x+3).
- 3^xln3.
- 5/(5x−1).
- e^x+xe^x=e^x(1+x).
- 2xcos x−x²sin x.
18. Transfer check
Before differentiating, name the outer structure and every inner function. After differentiating, ask whether the sign, domain and graph behaviour are plausible. This two-stage habit is more robust than relying on memory alone.
Continue with Derivative Rules, the Differentiation guide, or return to the Additional Mathematics Hub.