Derivatives of Transcendental Functions

Transcendental functions (like sine, cosine, exe^x, ln⁡x\ln x) are not algebraic; they "transcend" algebra. They are fundamental to modeling periodic phenomena, growth, decay, and many engineering applications.

Trigonometric Functions

The derivatives of sine and cosine are cyclic. The derivative of sine is cosine, and the derivative of cosine is negative sine.

Trigonometric Derivatives

  1. ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x
  2. ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x
  3. ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x
  4. ddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx}[\csc x] = -\csc x \cot x
  5. ddx[sec⁡x]=sec⁡xtan⁡x\frac{d}{dx}[\sec x] = \sec x \tan x
  6. ddx[cot⁡x]=−csc⁡2x\frac{d}{dx}[\cot x] = -\csc^2 x
Chain Rule Version: ddx[sin⁡u]=cos⁡u⋅u′\frac{d}{dx}[\sin u] = \cos u \cdot u'

Function vs Derivative

f(x) = x²
f(x)0.00
f'(x)0.00

Blue Line: The function f(x)f(x).

Red Line: The derivative f′(x)f'(x). Notice that when f(x)f(x) has a peak or valley, f′(x)=0f'(x) = 0.

Inverse Trigonometric Functions

The inverse trigonometric functions (arcsin⁡x\arcsin x, arccos⁡x\arccos x, etc.) have their own differentiation rules derived implicitly. Notice the domain restrictions (e.g., inside the square root must be positive).

Inverse Trig Derivatives

  1. ddx[arcsin⁡u]=u′1−u2\frac{d}{dx}[\arcsin u] = \frac{u'}{\sqrt{1-u^2}}
  2. ddx[arccos⁡u]=−u′1−u2\frac{d}{dx}[\arccos u] = -\frac{u'}{\sqrt{1-u^2}}
  3. ddx[arctan⁡u]=u′1+u2\frac{d}{dx}[\arctan u] = \frac{u'}{1+u^2}
  4. ddx[arccotu]=−u′1+u2\frac{d}{dx}[\text{arccot} u] = -\frac{u'}{1+u^2}
  5. ddx[arcsecu]=u′∣u∣u2−1\frac{d}{dx}[\text{arcsec} u] = \frac{u'}{|u|\sqrt{u^2-1}}
  6. ddx[arccscu]=−u′∣u∣u2−1\frac{d}{dx}[\text{arccsc} u] = -\frac{u'}{|u|\sqrt{u^2-1}}

Exponential and Logarithmic Functions

The exponential function exe^x is unique in calculus: it is its own derivative. This property makes it the natural choice for describing growth proportional to size. For other bases, we use the chain rule with a natural logarithm scaling factor.

Exponential & Logarithmic Rules

  1. Natural Exponential: ddx[eu]=eu⋅u′\frac{d}{dx}[e^u] = e^u \cdot u'
  2. General Exponential: ddx[au]=au(ln⁡a)⋅u′\frac{d}{dx}[a^u] = a^u (\ln a) \cdot u'. This introduces a constant scaling factor ln⁡a\ln a.
  3. Natural Logarithm: ddx[ln⁡u]=u′u\frac{d}{dx}[\ln u] = \frac{u'}{u}. This connects logarithmic functions to rational functions.
  4. General Logarithm: ddx[log⁡au]=u′uln⁡a\frac{d}{dx}[\log_a u] = \frac{u'}{u \ln a}. Using the change of base formula, this is essentially differentiating ln⁡uln⁡a\frac{\ln u}{\ln a}.

Exponential Growth: P(t)=P0ertP(t) = P_0 e^{rt}

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Observation: The solid blue line represents the population P(t)P(t), and the dashed red line is its derivative P′(t)P'(t). Notice how P′(t)P'(t) is always a constant multiple of P(t)P(t).

Logarithmic Differentiation

Logarithmic differentiation is a technique that uses properties of logarithms to simplify the differentiation of complex products, quotients, and powers, especially when the variable appears in both the base and the exponent (like xxx^x).

Steps for Logarithmic Differentiation

  1. Take the natural logarithm (ln⁡\ln) of both sides of the equation.
Use logarithm properties to simplify the expression:
  • ln⁡(AB)=ln⁡A+ln⁡B\ln(AB) = \ln A + \ln B
  • ln⁡(A/B)=ln⁡A−ln⁡B\ln(A/B) = \ln A - \ln B
  • ln⁡(AB)=Bln⁡A\ln(A^B) = B \ln A
  1. Differentiate implicitly with respect to xx. Remember the derivative of ln⁡y\ln y is 1ydydx\frac{1}{y}\frac{dy}{dx}.
  2. Solve for dydx\frac{dy}{dx} and substitute the original expression for yy.

Hyperbolic Functions

Hyperbolic functions are defined using exponentials (exe^x and e−xe^{-x}) and relate to hyperbolas similarly to how trig functions relate to circles. They appear frequently in engineering (e.g., the catenary curve of a hanging cable).

Hyperbolic Functions

  • sinh⁡x=ex−e−x2\sinh x = \frac{e^x - e^{-x}}{2}
  • cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2}
  • tanh⁡x=sinh⁡xcosh⁡x\tanh x = \frac{\sinh x}{\cosh x}

Hyperbolic Derivatives

  1. ddx[sinh⁡u]=cosh⁡u⋅u′\frac{d}{dx}[\sinh u] = \cosh u \cdot u'
  2. ddx[cosh⁡u]=sinh⁡u⋅u′\frac{d}{dx}[\cosh u] = \sinh u \cdot u' (Positive sign, unlike cosine!)
  3. ddx[tanh⁡u]=sech2u⋅u′\frac{d}{dx}[\tanh u] = \text{sech}^2 u \cdot u'

Inverse Hyperbolic Functions

The inverse hyperbolic functions also possess unique differentiation properties. These derivatives frequently lead to algebraic expressions containing inverse roots, proving incredibly useful for solving advanced integrals and differential equations.

Inverse Hyperbolic Derivatives

  1. ddx[arcsinh u]=u′u2+1\frac{d}{dx}[\text{arcsinh } u] = \frac{u'}{\sqrt{u^2+1}}
  2. ddx[arccosh u]=u′u2−1\frac{d}{dx}[\text{arccosh } u] = \frac{u'}{\sqrt{u^2-1}}
  3. ddx[arctanh u]=u′1−u2\frac{d}{dx}[\text{arctanh } u] = \frac{u'}{1-u^2}
  4. ddx[arccoth u]=u′1−u2\frac{d}{dx}[\text{arccoth } u] = \frac{u'}{1-u^2}
Key Takeaways
  • Trig Derivatives follow a cyclic pattern. Remember the negative signs for co-functions (cos⁡,cot⁡,csc⁡\cos, \cot, \csc).
  • Inverse Trig derivatives are algebraic functions involving roots and squares.
  • exe^x is the only function whose derivative is itself.
  • ln⁡x\ln x has a derivative of 1/x1/x, linking logarithms to rational functions.
  • Logarithmic differentiation simplifies taking the derivative of expressions with variables in exponents or complicated products/quotients.
  • Hyperbolic derivatives are very similar to trig derivatives but watch out for sign differences (e.g., derivative of cosh⁡\cosh is positive sinh⁡\sinh).
  • Inverse Hyperbolic derivatives yield rational functions or inverse square roots, distinguishing them fundamentally from their trigonometric counterparts.
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