Overview
This lecture reviews standard derivative formulas essential for calculus and engineering, focusing on polynomial, exponential, and logarithmic rules used for quickly finding derivatives.
The Concept of the Derivative
- The derivative represents the slope of a function, defined as the ratio of change in height to change in width as the interval becomes infinitesimally small.
- Calculating derivatives from first principles involves limits, but standard rules simplify the process for common functions.
Basic Derivative Rules
- The derivative of a constant is zero (e.g., d/dx [5] = 0).
- Power Rule: For d/dx [xⁿ], bring the power n down front and subtract 1 from the exponent (e.g., d/dx [x³] = 3x²).
- Sum and Difference Rule: The derivative of a sum or difference is the sum or difference of the derivatives (d/dx [f(x) ± g(x)] = f'(x) ± g'(x)).
- Constant Multiple Rule: A constant multiplied by a function stays as a multiplier in the derivative (d/dx [c·f(x)] = c·f'(x)).
Applying Rules to Examples
- Rewrite roots and fractions as powers to apply the power rule easily (e.g., √x = x¹ᐟ²).
- Simplify expressions by combining and tidying constants and exponents as part of standard practice.
Exponential Derivatives
- The derivative of eˣ is eˣ.
- The derivative of aˣ (where a is a constant) is aˣ times ln(a).
- The special property of eˣ is that its derivative is itself, due to the mathematical constant e.
Logarithmic Derivatives
- The derivative of ln(x) is 1/x.
- The derivative of log base a of x is 1/(x·ln(a)).
- Use these rules carefully, noting variable placement in the function.
Practice Examples and Key Strategies
- For expressions with both powers and constants, apply the correct rule based on whether the variable is in the exponent or the base.
- For mixed examples, use multipliers properly and simplify where possible.
Key Terms & Definitions
- Derivative — The instantaneous rate of change or slope of a function.
- Power Rule — d/dx [xⁿ] = n·xⁿ⁻¹.
- Exponential Rule — d/dx [aˣ] = aˣ·ln(a); d/dx [eˣ] = eˣ.
- Logarithmic Rule — d/dx [ln(x)] = 1/x; d/dx [logₐ(x)] = 1/(x·ln(a)).
- Constant Multiple Rule — d/dx [c·f(x)] = c·f'(x).
- Sum & Difference Rule — d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
Action Items / Next Steps
- Review textbook readings or the QN Prep site to reinforce these derivative rules.
- Practice rewriting roots and fractions as powers.
- Complete assigned problems using these standard derivative formulas.