Derivatives of Exponential and Inverse Trigonometric Functions

Derivatives of Exponential and Inverse Trigonometric Functions

1. Derivatives of Exponential Functions

a. Natural Exponential Function (𝑒ˣ)
The derivative of $e^x$ is unique because it is equal to itself:
$\frac{d}{dx} e^x = e^x$

b. General Exponential Functions (𝑎ˣ)
For any base $a > 0$, the derivative incorporates a logarithmic correction factor:
$\frac{d}{dx} a^x = a^x \ln a$
Special case: If $a = e$, then $\ln e = 1$, reducing to the natural exponential case.

c. Chain Rule Applications
When the exponent is a function $u(x)$:
$\frac{d}{dx} e^{u(x)} = e^{u(x)} \cdot u'(x)$
$\frac{d}{dx} a^{u(x)} = a^{u(x)} \ln a \cdot u'(x)$

Example:
$\frac{d}{dx} e^{3x^2} = e^{3x^2} \cdot 6x$

2. Derivatives of Inverse Trigonometric Functions

Each inverse trig function has a distinct derivative formula, often involving square roots in the denominator.

a. Arcsine (sin⁻¹𝑥)
$\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}} \quad \text{for } |x| < 1$

b. Arccosine (cos⁻¹𝑥)
$\frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}} \quad \text{for } |x| < 1$

c. Arctangent (tan⁻¹𝑥)
$\frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2} \quad \text{for all real } x$

d. Arcsecant (sec⁻¹𝑥)
$\frac{d}{dx} \sec^{-1} x = \frac{1}{|x| \sqrt{x^2 - 1}} \quad \text{for } |x| > 1$

e. Arccosecant (csc⁻¹𝑥)
$\frac{d}{dx} \csc^{-1} x = -\frac{1}{|x| \sqrt{x^2 - 1}} \quad \text{for } |x| > 1$

f. Arccotangent (cot⁻¹𝑥)
$\frac{d}{dx} \cot^{-1} x = -\frac{1}{1 + x^2} \quad \text{for all real } x$

3. Proof Techniques

a. Exponential Derivatives

• Use the limit definition of the derivative and properties of $e^h$.
• For $a^x$, rewrite as $e^{x \ln a}$ and apply the chain rule.

b. Inverse Trig Derivatives

• Implicit differentiation:
  Example for $\sin^{-1} x$:
  Let $y = \sin^{-1} x$. Then $x = \sin y$.
  Differentiate implicitly: $1 = \cos y \cdot \frac{dy}{dx}$.
  Solve for $\frac{dy}{dx}$, using $\cos y = \sqrt{1 - x^2}$.

4. Common Mistakes to Avoid

1. Chain Rule Errors: Forgetting to multiply by $u'(x)$ in $e^{u(x)}$ or $\sin^{-1}(u(x))$.
   Incorrect: $\frac{d}{dx} e^{2x} = e^{2x}$
   Correct: $\frac{d}{dx} e^{2x} = 2e^{2x}$.

2. Domain Restrictions:

   • $\frac{d}{dx} \sin^{-1} x$ is undefined for $|x| \geq 1$.
   • $\frac{d}{dx} \sec^{-1} x$ requires $|x| > 1$.

3. Sign Errors:

   • Confusing $\cos^{-1} x$ and $\sin^{-1} x$ derivatives (negative sign).
   • Misapplying derivatives of $\cot^{-1} x$ vs. $\tan^{-1} x$.

5. Applications

• Exponential Growth/Decay: Modeling populations or radioactive decay.
• Physics: Wave equations involving $e^{i\omega t}$ (complex exponentials).
• Engineering: Control systems with inverse trig functions (e.g., phase analysis).

6. Practice Problems

1. Compute $\frac{d}{dx} \left( e^{x^3} + \tan^{-1}(5x) \right)$.
   Solution: $3x^2 e^{x^3} + \frac{5}{1 + 25x^2}$.

2. Find $\frac{d}{dx} \sec^{-1}(e^x)$.
   Solution: $\frac{e^x}{e^x \sqrt{e^{2x} - 1}} = \frac{1}{\sqrt{e^{2x} - 1}}$.

3. Differentiate $y = 2^{\sin x}$.
   Solution: $y' = 2^{\sin x} \ln 2 \cdot \cos x$.

7. Key Takeaways

• Exponentials: The derivative of $e^x$ is itself; other bases require $\ln a$.
• Inverse Trig: Memorize the six derivative forms and their domains.
• Chain Rule: Critical for composite functions (e.g., $e^{u(x)}$, $\tan^{-1}(u(x))$).

Mastery of these derivatives is essential for calculus, differential equations, and advanced applied mathematics.