Derivatives: Hyperbolic Functions & L'Hospital's Rule

TAKEAWAYS:

• Hyperbolic functions are defined using exponentials and behave like trigonometric functions but for hyperbolas.
• Derivatives of sinh x = cosh x, cosh x = sinh x, and tanh x = sech² x (memorize these!).
• L'Hospital's Rule replaces indeterminate limits with derivatives of numerator and denominator.
• Always check for indeterminate forms (0/0 or ∞/∞) before applying L'Hospital's Rule.
• The Rule can be applied repeatedly if the limit remains indeterminate after one application.
• Common mistakes include forgetting to check for indeterminate forms or misapplying the Rule to non-indeterminate cases.

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Hyperbolic Functions: Definitions and Graphs

Hyperbolic functions are defined using exponential functions and are analogous to trigonometric functions but for hyperbolas. The key hyperbolic functions are:

• sinh x (hyperbolic sine): $\sinh x=\frac{e^x-e^{-x}}{2}$
• cosh x (hyperbolic cosine): $\cosh x=\frac{e^x+e^{-x}}{2}$
• tanh x (hyperbolic tangent): $\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$

Graphs of Hyperbolic Functions

Key Observations:

• sinh x is an odd function (symmetric about the origin).
• cosh x is an even function (symmetric about the y-axis).
• tanh x is also an odd function and is bounded between -1 and 1.

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Derivatives of Hyperbolic Functions

The derivatives of hyperbolic functions are derived using their definitions and the rules of differentiation. Here are the key results:

1. Derivative of sinh x: $$\frac{d}{dx}(\sinh x)=\cosh x$$

2. Derivative of cosh x: $$\frac{d}{dx}(\cosh x)=\sinh x$$

3. Derivative of tanh x: $$\frac{d}{dx}(\tanh x)={\text{sech}}^{2}x$$ where $\text{sech}x=\frac{1}{\cosh x}$.

Worked Example 1: Differentiate $y=\sinh (3x)$

Solution: Using the chain rule: $$\frac{dy}{dx}=\cosh (3x)\cdot \frac{d}{dx}(3x)=3\cosh (3x)$$

Worked Example 2: Differentiate $y=x\cosh x-\sinh x$

Solution: Differentiate term by term: $$\frac{dy}{dx}=\cosh x+x\sinh x-\cosh x=x\sinh x$$

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L'Hospital's Rule: Evaluating Indeterminate Limits

L'Hospital's Rule is used to evaluate limits of the form $\frac{0}{0}$ or $\frac{\infty }{\infty }$. The Rule states:

If ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty }{\infty }$, then: $$>\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}>$$ provided the limit on the right exists.

When to Apply L'Hospital's Rule?

L'Hospital's Rule can only be applied if the limit is in one of the following indeterminate forms:

1. $\frac{0}{0}$
2. $\frac{\infty }{\infty }$

Example of Non-Indeterminate Forms (Do Not Apply L'Hospital's Rule):

• $\frac{5}{0}$ (undefined, not indeterminate)
• $\frac{3}{4}$ (determinate form)

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Steps to Apply L'Hospital's Rule

1. Check the form: Ensure the limit is $\frac{0}{0}$ or $\frac{\infty }{\infty }$.
2. Differentiate numerator and denominator: Find $f^{\prime }(x)$ and $g^{\prime }(x)$.
3. Evaluate the new limit: Compute ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$.
4. Repeat if necessary: If the new limit is still indeterminate, apply L'Hospital's Rule again.

Worked Example 3: Evaluate ${\lim }_{x\to 0}\frac{\sin x}{x}$

Solution:

1. Direct substitution gives $\frac{0}{0}$ (indeterminate).
2. Differentiate numerator and denominator: $$\frac{d}{dx}(\sin x)=\cos x,\frac{d}{dx}(x)=1$$
3. Apply L'Hospital's Rule: $$\underset{x\to 0}{\lim }\frac{\sin x}{x}=\underset{x\to 0}{\lim }\frac{\cos x}{1}=\cos 0=1$$

Worked Example 4: Evaluate ${\lim }_{x\to \infty }\frac{e^x}{x^2}$

Solution:

1. Direct substitution gives $\frac{\infty }{\infty }$ (indeterminate).
2. Differentiate numerator and denominator: $$\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(x^2)=2x$$
3. Apply L'Hospital's Rule: $$\underset{x\to \infty }{\lim }\frac{e^x}{x^2}=\underset{x\to \infty }{\lim }\frac{e^x}{2x}$$ The new limit is still $\frac{\infty }{\infty }$, so apply L'Hospital's Rule again: $$\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(2x)=2$$ $$\underset{x\to \infty }{\lim }\frac{e^x}{2x}=\underset{x\to \infty }{\lim }\frac{e^x}{2}=\infty$$ Thus, the original limit is $\infty$.

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Comparison Table: Hyperbolic vs. Trigonometric Functions

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Applications of Hyperbolic Functions and L'Hospital's Rule

1. Hyperbolic Functions:

   • Used in relativity (Lorentz transformations).
   • Model catenaries (shape of hanging cables).
   • Appear in differential equations (e.g., heat equation).

2. L'Hospital's Rule:

   • Evaluates limits in calculus (e.g., ${\lim }_{x\to 0}\frac{\sin x}{x}$).
   • Used in engineering (signal processing, control systems).
   • Helps in asymptotic analysis (behavior of functions at infinity).

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Common Mistakes to Avoid

1. Applying L'Hospital's Rule to non-indeterminate forms:

   • Example: ${\lim }_{x\to 2}\frac{x^2}{x+1}$ is not indeterminate, so do not apply the Rule.

2. Forgetting to check the form:

   • Always verify if the limit is $\frac{0}{0}$ or $\frac{\infty }{\infty }$ before applying the Rule.

3. Misapplying the Rule to $0\times \infty$ or $\infty -\infty$:

   • These forms require algebraic manipulation (e.g., combining fractions) before applying L'Hospital's Rule.

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Exam Tip

1. Memorize derivatives of hyperbolic functions:

   • $\frac{d}{dx}\sinh x=\cosh x$
   • $\frac{d}{dx}\cosh x=\sinh x$
   • $\frac{d}{dx}\tanh x={\text{sech}}^{2}x$

2. Always check for indeterminate forms:

   • Write "indeterminate" or "0/0" in your exam paper before applying L'Hospital's Rule.

3. Practice limit problems:

   • NEB often tests limits requiring L'Hospital's Rule, especially with trigonometric, exponential, or logarithmic functions.

4. Show all steps:

   • Write down each differentiation step clearly. Partial credit is given for correct intermediate steps.

5. Watch for repeated applications:

   • Some limits require applying L'Hospital's Rule more than once (e.g., $\frac{e^x}{x^2}$).

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NEB Board-Style Questions

Short Answer Questions (2 marks each)

1. Differentiate $y=\cosh (5x)$.
2. Evaluate ${\lim }_{x\to 0}\frac{e^x-1}{x}$ using L'Hospital's Rule.
3. State whether ${\lim }_{x\to \infty }\frac{\ln x}{x}$ is determinate or indeterminate. If indeterminate, apply L'Hospital's Rule to evaluate it.

Long Answer Questions (5-7 marks each)

4. Differentiate the following functions: a) $y=x^2\sinh x$ b) $y=\tanh (3x)$

5. Evaluate the following limits using L'Hospital's Rule: a) ${\lim }_{x\to 0}\frac{1-\cos x}{x^2}$ b) ${\lim }_{x\to \infty }\frac{x}{\sqrt{x^2+1}}$

6. Prove that $\frac{d}{dx}(\text{sech}x)=-\text{sech}x\tanh x$.

Problem-Solving (7-10 marks)

7. A particle moves along a curve defined by $y=\sinh x$. Find the slope of the tangent to the curve at $x=1$. Also, evaluate ${\lim }_{x\to \infty }\frac{\sinh x}{e^x}$.

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Note: Always practice writing clear, step-by-step solutions in your exam. NEB values methodical approaches over rushed answers!