The Ultimate Guide to Derivative Rules: Product, Quotient, and Chain Rule
Derivatives are a cornerstone of calculus, used to analyze rates of change, slopes of curves, and dynamic systems. This guide explains three essential derivative rules: the Product Rule, the Quotient Rule, and the Chain Rule. Whether you're a student, teacher, or professional revisiting calculus, this detailed guide is designed to clarify these concepts with simple and complex examples.
1. Product Rule
Definition
If u(x) and v(x) are two differentiable functions, the derivative of their product is:
\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]
Step-by-Step Explanation
- Identify the two functions \( u(x) \) and \( v(x) \) being multiplied.
- Compute the derivative of \( u(x) \), denoted \( u'(x) \).
- Compute the derivative of \( v(x) \), denoted \( v'(x) \).
- Apply the formula: Multiply \( u'(x) \) by \( v(x) \), and \( u(x) \) by \( v'(x) \).
- Add the two products to find the final derivative.
Example 1: Simple Case
Find the derivative of \( f(x) = x^2 \sin(x) \).
Solution:
- \( u(x) = x^2 \), so \( u'(x) = 2x \).
- \( v(x) = \sin(x) \), so \( v'(x) = \cos(x) \).
- Apply the Product Rule:
- Final answer:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) = (2x)(\sin(x)) + (x^2)(\cos(x)) \]
\[ f'(x) = 2x\sin(x) + x^2\cos(x) \]
Example 2: Complex Case
Find the derivative of \( f(x) = (3x^3 + 5x)e^x \).
Solution:
- \( u(x) = 3x^3 + 5x \), so \( u'(x) = 9x^2 + 5 \).
- \( v(x) = e^x \), so \( v'(x) = e^x \).
- Apply the Product Rule:
- Simplify:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) = (9x^2 + 5)(e^x) + (3x^3 + 5x)(e^x) \]
\[ f'(x) = e^x(3x^3 + 9x^2 + 5x + 5) \]
2. Quotient Rule
Definition
To differentiate the division of two functions:
\[ \frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}, \quad v(x) \neq 0 \]
Step-by-Step Explanation
- Identify the numerator \( u(x) \) and the denominator \( v(x) \).
- Compute \( u'(x) \) and \( v'(x) \).
- Apply the formula: Multiply \( v(x) \) by \( u'(x) \), then subtract the product of \( u(x) \) and \( v'(x) \).
- Divide the result by \( [v(x)]^2 \).
Example 1: Simple Case
Find the derivative of \( f(x) = \frac{x^2}{\cos(x)} \).
Solution:
- \( u(x) = x^2 \), so \( u'(x) = 2x \).
- \( v(x) = \cos(x) \), so \( v'(x) = -\sin(x) \).
- Apply the Quotient Rule:
- Simplify:
\[ f'(x) = \frac{(\cos(x))(2x) - (x^2)(-\sin(x))}{\cos^2(x)} \]
\[ f'(x) = \frac{2x\cos(x) + x^2\sin(x)}{\cos^2(x)} \]
3. Chain Rule
Definition
The Chain Rule helps differentiate composite functions:
\[ \frac{dy}{dx} = g'(f(x)) \cdot f'(x) \]
Example: Complex Case
Find the derivative of \( f(x) = \sin(e^{x^2}) \).
Solution:
- Outer function: \( g(u) = \sin(u) \), so \( g'(u) = \cos(u) \).
- Inner function: \( u = e^{x^2} \), so \( u' = e^{x^2} \cdot 2x \).
- Apply the Chain Rule:
- Final answer:
\[ f'(x) = \cos(e^{x^2}) \cdot (e^{x^2} \cdot 2x) \]
\[ f'(x) = 2xe^{x^2}\cos(e^{x^2}) \]
Mastering these rules is essential for success in calculus. Practice them with a variety of examples to build your confidence and deepen your understanding.