Rules for Differentiation

Differentiation is a fundamental concept in calculus that allows us to determine how a function changes at any point. With that in mind, let's delve into the basic rules of differentiation that are essential for understanding and applying this powerful mathematical tool.

1. Power Rule

The Power Rule is one of the simplest and most commonly used rules for differentiation. It states that if you have a function of the form:

\[ f(x) = x^n \]

where \( n \) is any real number, then the derivative of \( f \) with respect to \( x \) is:

\[ f'(x) = nx^{n-1} \]

Example:

To illustrate the Power Rule, let's differentiate the function \( f(x) = x^3 \):

  1. Identify \( n \): Here, \( n = 3 \).

  2. Apply the Power Rule:

    \[ f'(x) = 3x^{3-1} = 3x^2 \]

So, the derivative of \( f(x) = x^3 \) is \( f'(x) = 3x^2 \).

2. Product Rule

The Product Rule is used when you need to differentiate the product of two functions. If you have two differentiable functions \( u(x) \) and \( v(x) \), the derivative of their product is given by:

\[ (uv)' = u'v + uv' \]

Example:

Consider \( f(x) = x^2 \sin(x) \). Here, let \( u = x^2 \) and \( v = \sin(x) \).

  1. Differentiate \( u \): \( u' = 2x \)

  2. Differentiate \( v \): \( v' = \cos(x) \)

  3. Apply the Product Rule:

    \[ f'(x) = u'v + uv' = (2x) \sin(x) + (x^2) \cos(x) \]

Therefore, the derivative \( f'(x) = 2x\sin(x) + x^2\cos(x) \).

3. Quotient Rule

The Quotient Rule comes into play when you are differentiating a function that is the quotient of two other functions. If you have \( u(x) \) and \( v(x) \) as two differentiable functions, then the quotient \( \frac{u}{v} \) is differentiated as follows:

\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \]

Example:

Let's differentiate \( f(x) = \frac{x^2}{\cos(x)} \). Here, \( u = x^2 \) and \( v = \cos(x) \).

  1. Differentiate \( u \): \( u' = 2x \)

  2. Differentiate \( v \): \( v' = -\sin(x) \)

  3. Apply the Quotient Rule:

    \[ f'(x) = \frac{u'v - uv'}{v^2} = \frac{(2x)(\cos(x)) - (x^2)(-\sin(x))}{(\cos(x))^2} \]

Therefore, the derivative \( f'(x) = \frac{2x\cos(x) + x^2\sin(x)}{\cos^2(x)} \).

4. Chain Rule

The Chain Rule is a powerful rule used to differentiate composite functions. If \( f(g(x)) \) is a composite function, then the derivative is given by:

\[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \]

Example:

Consider \( f(x) = \sin(x^2) \). We can identify \( g(x) = x^2 \) and \( f(g) = \sin(g) \).

  1. Differentiate \( f(g) \): \( f'(g) = \cos(g) \)

  2. Differentiate \( g(x) \): \( g'(x) = 2x \)

  3. Apply the Chain Rule:

    \[ f'(x) = \cos(g(x)) \cdot g'(x) = \cos(x^2) \cdot (2x) = 2x\cos(x^2) \]

Thus, the derivative of \( f(x) = \sin(x^2) \) is \( f'(x) = 2x\cos(x^2) \).

5. Higher-Order Derivatives

Once you are comfortable with finding the first derivative, you can extend your work to find higher-order derivatives. The second derivative is simply the derivative of the first derivative:

\[ f''(x) = (f'(x))' \]

You can continue this process to find the third derivative \( f'''(x) \), and so on.

Example:

If we start with \( f(x) = x^3 \) and found that \( f'(x) = 3x^2 \), we can find the second derivative:

  1. Differentiate \( f'(x) \):

    \[ f''(x) = (3x^2)' = 6x \]

  2. Differentiate \( f''(x) \):

    \[ f'''(x) = (6x)' = 6 \]

So, for \( f(x) = x^3 \), we found:

  • \( f'(x) = 3x^2 \)
  • \( f''(x) = 6x \)
  • \( f'''(x) = 6 \)

6. Special Derivatives

Some functions have specific rules that apply to their derivatives. These include:

  • The derivative of a constant \( c \):

    \[ \frac{d}{dx}(c) = 0 \]

  • The derivative of \( \sin(x) \):

    \[ \frac{d}{dx}(\sin(x)) = \cos(x) \]

  • The derivative of \( \cos(x) \):

    \[ \frac{d}{dx}(\cos(x)) = -\sin(x) \]

  • The derivative of \( e^x \):

    \[ \frac{d}{dx}(e^x) = e^x \]

  • The derivative of \( \ln(x) \):

    \[ \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]

These rules are handy shortcuts that simplify the differentiation process for specific functions.

Conclusion

Understanding these basic rules of differentiation—Power Rule, Product Rule, Quotient Rule, and Chain Rule—forms the foundation for tackling more complex calculus problems. With practice, you'll find that differentiating functions becomes second nature, allowing you to analyze and interpret the behavior of various mathematical models effectively. Happy differentiating!