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Basic Differentiation Rules

·494 words·3 mins
Author
Adrian

Basic Differentiation Rules
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The power rule allows one to differentiate any polynomial with the following formula, where \( n \) is a constant:

\[ \frac{d}{dx} x^n = n x^{n-1} \]

For example, the derivative of \( x^2 \) is \( 2x \) because

\[ \frac{d}{dx} x^2 = 2x^{2-1} = 2x \]

Positive Integer Powers
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Using the power rule, evaluate the following:

  • What is the derivative of \( x \)?
  • What is the derivative of \( x^3 \)?
  • What is the derivative of \( x + x^2 + x^3 \)?
  • What is the derivative of \( x + 2x^2 + 3x^3 \)?
  • What is the derivative of \( 3x + 3x^2 + 3x^3 \)?

Fractional and Negative Powers
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The power rule also applies to fractional and negative exponents. To differentiate \( \sqrt{x} \), rewrite it as \( x^{1/2} \) and apply the rule:

\[ \begin{aligned} \frac{d}{dx} \sqrt{x} &= \frac{d}{dx} x^{1/2} \\ &= \frac{1}{2} x^{-1/2} \\ &= \frac{1}{2\sqrt{x}} = \frac{2\sqrt{x}}{4x} \end{aligned} \]

So the derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \) or \( \frac{2\sqrt{x}}{4x} \).

Let’s try \( \frac{1}{x} \):

\[ \begin{aligned} \frac{d}{dx} \frac{1}{x} &= \frac{d}{dx} x^{-1} \\ &= -x^{-2} \\ &= -\frac{1}{x^2} \end{aligned} \]

So the derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

Exercises
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  • If \( f(x) = \frac{2}{x^2} \), what is \( f'(x) \)?
  • What is the derivative of \( 7x^{-7} + 3x \)?
  • Prove the following: \( \frac{d}{dx} nx = n \)

Differentiating Exponential Functions
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The Immortal Exponential Function
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The derivative of \( e^x \) is \( e^x \), satisfying the differential equation \( f(x) = f'(x) \). Its derivative remains the same no matter how many times you differentiate.

  • What is the 72,934th derivative of \( e^x \)?
  • Simplify: \( \sum_{n=1}^{1000} f^{(n)}(x) \), given \( f(x) = e^x \)

The Product Rule
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To differentiate a product of functions, use the product rule:

\[ \frac{d}{dx} \big(f(x)g(x)\big) = f'(x)g(x) + f(x)g'(x) \]

Example:

\[ \begin{aligned} \frac{d}{dx} \big(x^3 \cos(x)\big) &= (\frac{d}{dx} x^3)\cos(x) + x^3(\frac{d}{dx} \cos(x)) \\ &= 3x^2 \cos(x) - x^3 \sin(x) \end{aligned} \]

Exercises
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  • What is the derivative of \( \cos(x) \sin(x) \)?
  • What is the derivative of \( e^x x^2 \)?
  • What is the derivative of \( e^x \sin(x) + 2x^2 \cos(x) \)?

The Quotient Rule
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Coming soon…

Differentiating Trigonometric Functions
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Here are some key derivatives of basic trigonometric functions:

\[ \begin{aligned} \frac{d}{dx} \sin(x) &= \cos(x) \\ \frac{d}{dx} \cos(x) &= -\sin(x) \\ \frac{d}{dx} [-\sin(x)] &= -\cos(x) \\ \frac{d}{dx} [-\cos(x)] &= \sin(x) \end{aligned} \]

Exercises
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  • What is \( f'(x) \) if \( f(x) = \sin(x) \)?
  • What is \( f''(x) \) if \( f(x) = \sin(x) \)?
  • What is \( f^{(3)}(x) \) if \( f(x) = \sin(x) \)?
  • What is \( f^{(4)}(x) \) if \( f(x) = \sin(x) \)?
  • What is \( f^{(5)}(x) \) if \( f(x) = \sin(x) \)?
  • What is \( f^{(400)}(x) \) if \( f(x) = \sin(x) \)?
  • Give one solution to the differential equation \( f(x) = -f''(x) \)