#Calculating Derivatives: Comprehensive Study Notes

#Table of Contents

1. Introduction to Derivatives
2. Estimating Derivatives
3. Finding the Derivative as an Expression
   • Basic Differentiation
   • Differentiating Exponentials and Logarithms
   • Differentiating Trigonometric Functions
   • Product Rule and Quotient Rule
   • Chain Rule
   • Inverse Function Theorem
   • Implicit Differentiation
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

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#Introduction to Derivatives

A derivative of a function represents the rate at which the function's value changes as its input changes. The derivative of $f(x)$ is defined as: $${f}'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

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#Estimating Derivatives

To estimate a derivative at a point using a table or graph, remember:

• The derivative at a point equals the slope of the tangent at that point.
• To approximate the slope of the tangent to the graph of $f(x)$ at $x = a$:
  • Find the slope of line segments joining nearby coordinates that lie on the graph.

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#Finding the Derivative as an Expression

#Basic Differentiation

1. Powers of $x$:
   • If $f(x) = x^n$, then $f'(x) = nx^{n-1}$
2. Sums and Differences:
   • The derivative of a sum (or difference) of terms is the sum (or difference) of the derivatives of the terms.
3. Constant Multiples:
   • If $f(x) = ax^n$, then $f'(x) = an x^{n-1}$
4. Special Cases:
   • If $f(x) = ax$, then $f'(x) = a$
   • If $g(x) = a$, then $g'(x) = 0$

#Differentiating Exponentials and Logarithms

#Differentiating Trigonometric Functions

#Product Rule and Quotient Rule

1. Product Rule:

   • If $f(x) = g(x) \cdot h(x)$, then: $$f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$$
   • Alternatively, if $y = u \cdot v$, then: $$\frac{dy}{dx} = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx}$$ $$y' = u'v + uv'$$

2. Quotient Rule:

   • If $f(x) = \frac{g(x)}{h(x)}$, then: $$f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2}$$
   • Alternatively, if $y = \frac{u}{v}$, then: $$\frac{dy}{dx} = \frac{\frac{du}{dx} \cdot v - u \cdot \frac{dv}{dx}}{v^2}$$ $$y' = \frac{u'v - uv'}{v^2}$$

#Chain Rule

1. Chain Rule:
   • If $y = f(u)$ and $u = g(x)$, then: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
   • In function notation, if $h(x) = f(g(x))$, then: $$h'(x) = f'(g(x)) \cdot g'(x)$$

#Inverse Function Theorem

1. Inverse Function Theorem:
   • For a function $f$, the derivative of its inverse is given by: $$\left( f^{-1} \right)'(a) = \frac{1}{f'(f^{-1}(a))}$$
   • Alternatively, if $g(a) = f^{-1}(a)$: $$g'(a) = \frac{1}{f'(g(a))}$$
   • If $y = f^{-1}(x)$ so that $x = f(y)$, then: $$\frac{dy}{dx} = \frac{1}{\left( \frac{dx}{dy} \right)}$$

#Implicit Differentiation

1. Implicit Differentiation:
   • For equations like $3x^2 - 7xy^2 = 3$ or $x^2 + y^2 = 25$:
     • Every term in the equation is differentiated.
     • For terms involving $y$, apply the chain rule: $$\frac{d}{dx} f(y) = f'(y) \cdot y' = f'(y) \cdot \frac{dy}{dx}$$

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#Practice Questions

Practice Question

1. Differentiate $f(x) = 3x^4 - 5x^2 + 2x - 7$.

Practice Question

2. Find the derivative of $g(x) = e^{2x}$.

Practice Question

3. Use the product rule to differentiate $h(x) = (3x^2)(\sin x)$.

Practice Question

4. Apply the chain rule to differentiate $y = \sin(3x^2)$.

Practice Question

5. Differentiate implicitly: $x^2 + y^2 = 1$.

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#Glossary

• Derivative: The rate of change of a function with respect to its variable.
• Tangent: A line that touches a curve at a point without crossing over.
• Product Rule: A rule for differentiating products of two functions.
• Quotient Rule: A rule for differentiating quotients of two functions.
• Chain Rule: A rule for differentiating composite functions.
• Implicit Differentiation: A method for differentiating functions that are not explicitly solved for one variable.

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#Summary and Key Takeaways

• Understand the various methods for differentiating functions.
• Use the definition of the derivative only if explicitly asked.
• Differentiate basic functions, exponentials, logarithms, and trigonometric functions using their respective rules.
• Apply the product rule, quotient rule, and chain rule where appropriate.
• Use the inverse function theorem and implicit differentiation for more complex functions.
• Practice these techniques regularly to build confidence and proficiency.

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By mastering these concepts, students will be well-prepared to tackle a variety of differentiation problems in their exams.