Differentiating Special Functions (DP IB Applications & Interpretation (AI): HL): Revision Note

Differentiating trig functions

How do I differentiate sin, cos and tan?

• The derivative of$\mathit{y}=\mathbf{sin}\mathit{x}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}=\mathbf{cos}\mathit{x}$   

• The derivative of$\mathit{y}=\mathbf{cos}\mathit{x}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}=-\mathbf{sin}\mathit{x}$

• The derivative of$\mathit{y}\mathbf{=}\mathbf{tan}\mathit{x}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{cos}}^{\mathbf{2}}\mathit{x}}$

  • This result for $\tan x$can be derived using quotient rule

• For the linear function$\mathit{a}\mathit{x}\mathbf{+}\mathit{b}$, where $\mathit{a}$ and $\mathit{b}$ are constants,

  • the derivative of$\mathit{y}\mathbf{=}\mathbf{sin}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\mathit{a}\mathbf{cos}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}$ 

  • the derivative of$\mathit{y}\mathbf{=}\mathbf{cos}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\mathbf{-}\mathit{a}\mathbf{sin}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}$

  • the derivative of$\mathit{y}\mathbf{=}\mathbf{tan}\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{\mathit{a}}{{\mathbf{cos}}^{\mathbf{2}}\mathbf{(}\mathit{a}\mathit{x}\mathit{+}\mathit{b}\mathbf{)}}$

• For the general function$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$,

  • the derivative of$\mathit{y}=\mathbf{sin}\left(\mathit{f}\left(\mathit{x}\right)\right)$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{cos}\left(\mathit{f}\left(x\right)\right)$

  • the derivative of$\mathit{y}=\mathbf{cos}\left(\mathit{f}\left(\mathit{x}\right)\right)$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\mathbf{-}{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{sin}\left(\mathit{f}\left(x\right)\right)$

  • the derivative of$\mathit{y}\mathbf{=}\mathbf{tan}\mathbf{\left(}\mathit{f}\mathbf{\right(}\mathit{x}\mathbf{\left)}\mathbf{\right)}$ is$\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}}{{\mathbf{cos}}^{\mathbf{2}}\mathbf{\left(}\mathit{f}\mathbf{\right(}\mathit{x}\mathbf{\left)}\mathbf{\right)}}$

• These last three results can be derived using the chain rule

• For calculus with trigonometric functions angles must be measured in radians

  • Ensure you know how to change the angle mode on your GDC

Differentiating e^x & lnx

How do I differentiate exponentials and logarithms?

• The derivative of $\mathit{y}\mathbf{=}{\mathit{e}}^{\mathit{x}}$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}{\mathit{e}}^{\mathit{x}}$ where $x\in \mathrm{ℝ}$

• The derivative of $\mathit{y}\mathbf{=}\mathbf{ln}\mathit{x}$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{\mathbf{1}}{\mathit{x}}$ where $x>0$

• For the linear function $\mathit{a}\mathit{x}\mathbf{+}\mathit{b}$, where $a$ and $b$ are constants,

  • the derivative of $\mathit{y}\mathbf{=}{\mathbf{e}}^{\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}}$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\mathit{a}{\mathbf{e}}^{\mathbf{(}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}\mathbf{)}}$

  • the derivative of $\mathit{y}=\mathbf{ln}\left(\mathit{a}\mathit{x}+\mathit{b}\right)$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}=\frac{\mathit{a}}{\left(\mathit{a}\mathit{x}+\mathit{b}\right)}$

    • in the special case $b=0$, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x}$     ($a$'s cancel)

• For the general function $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$,

  • the derivative of $\mathit{y}\mathbf{=}{\mathbf{e}}^{\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}}$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}{\mathbf{e}}^{\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}}$

  • the derivative of $\mathit{y}\mathbf{=}\mathbf{ln}\mathbf{\left(}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{\right)}$ is $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{{\mathit{f}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathit{x}\mathbf{\right)}}{\mathit{f}\left(\mathit{x}\right)}$

• The last two sets of results can be derived using the chain rule