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CIE A-Level Maths Study Notes

2.4.1 Advanced Differentiation Techniques

Differentiation, a critical tool in calculus, allows us to calculate the rate at which a function is changing at any given point. Advanced differentiation techniques are essential for tackling more complex functions and mathematical problems.

Basic Rules of Differentiation

  • Power Rule
    The derivative of xnx^n is nxn1nx^{n-1}.
    Example: f(x)=x3f(x) = x^3
    Solution: f(x)=3x2f'(x) = 3x^2
  • Exponential Rule
    The derivative of eue^u is dudxeu\frac{du}{dx}e^u.
    Example: f(x)=e2xf(x) = e^{2x}
    Solution: f(x)=2e2xf'(x) = 2e^{2x}
  • Logarithmic Rule ln(u)\ln(u) is dudx1u\frac{du}{dx}\frac{1}{u}.
    Example: f(x)=ln(5x)f(x) = \ln(5x)
    Solution: f(x)=1xf'(x) = \frac{1}{x}
  • Trigonometric Rules
    • The derivative of sin(ax)\sin(ax) is acos(ax)a\cos(ax).
      Example: f(x)=sin(3x)f(x) = \sin(3x)
      Solution: f(x)=3cos(3x)f'(x) = 3\cos(3x)
    • The derivative of cos(ax)\cos(ax) is asin(ax)-a\sin(ax).
      Example: f(x)=cos(4x)f(x) = \cos(4x)
      Solution: f(x)=4sin(4x)f'(x) = -4\sin(4x)
    • The derivative of tan(ax)\tan(ax) is asec2(ax)a\sec^2(ax).
      Example: f(x)=tan(2x)f(x) = \tan(2x)
      Solution: f(x)=2sec2(2x)f'(x) = 2\sec^2(2x)
  • Inverse Trigonometric Rule
    The derivative of tan1(ax)\tan^{-1}(ax) is a1+(ax)2\frac{a}{1+(ax)^2}.
    Example: f(x)=tan1(3x)f(x) = \tan^{-1}(3x)
    Solution: f(x)=31+(3x)2f'(x) = \frac{3}{1+(3x)^2}

Differentiation of Algebraic Expressions

  • Constant Multiple Rule
    The derivative of kf(x)kf(x) is kf(x)kf'(x).
    Example: f(x)=7cos(x)f(x) = 7\cos(x)
    Solution: f(x)=7sin(x)f'(x) = -7\sin(x)
  • Sum and Difference Rule
    The derivative of f(x)±g(x)f(x) \pm g(x) is f(x)±g(x)f'(x) \pm g'(x).
    Example: f(x)=exx2f(x) = e^{x} - x^2
    Solution: f((x)=ex2xf'((x) = e^{x} - 2x

Differentiation of Composite Functions

  • Chain Rule
    If h(x)=f(g(x))h(x) = f(g(x)), then h(x)h'(x) is f(g(x))g(x)f'(g(x)) \cdot g'(x).
    Example: f(x=ln(sin(x))f(x = \ln(\sin(x))
    Solution: f(x)=cot(x)f'(x) = \cot(x)
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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