Basic Derivative Rules

Finding derivatives using the formal limit definition can be long and tedious. Fortunately, there are several shortcut rules that make calculating derivatives fast and straightforward.

The Power Rule

The Power Rule is the most common shortcut for polynomials and terms with exponents. To find the derivative of $x$ raised to any power $n$, bring the exponent down to the front and subtract $1$ from the original exponent:

$\frac{d}{dx} (x^n) = n x^{n-1}$

Constant, Constant Multiple, and Sum/Difference Rules

These rules tell us how to handle numbers and multiple terms in an expression.

• Constant Rule: The derivative of any constant number $c$ is zero. $\frac{d}{dx} (c) = 0$
• Constant Multiple Rule: If a function is multiplied by a constant $c$, you can pull the constant out and multiply it by the derivative of the function. $\frac{d}{dx} [c f(x)] = c \cdot f'(x)$
• Sum and Difference Rules: The derivative of a sum or difference is simply the sum or difference of the individual derivatives. $\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$

Exponential and Logarithmic Functions

Exponential and logarithmic functions have their own unique derivative rules:

• Natural Exponential Function: The derivative of $e^x$ is just itself! $\frac{d}{dx} (e^x) = e^x$
• Natural Logarithm: The derivative of $\ln x$ is $1$ over $x$. $\frac{d}{dx} (\ln x) = \frac{1}{x}$

Basic Trigonometric Functions

You will frequently encounter sine and cosine in calculus. Memorize these two fundamental rules:

• Derivative of Sine: $\frac{d}{dx} (\sin x) = \cos x$
• Derivative of Cosine: $\frac{d}{dx} (\cos x) = -\sin x$

Example Problems

Example 1: Find $\frac{d}{dx}(3x^4 - 2x^2 + 5x - 7)$

Using the Sum/Difference, Constant Multiple, and Power Rules, we take the derivative of each term one by one:

• $\frac{d}{dx}(3x^4) = 3 \cdot 4x^3 = 12x^3$
• $\frac{d}{dx}(-2x^2) = -2 \cdot 2x^1 = -4x$
• $\frac{d}{dx}(5x) = 5 \cdot 1x^0 = 5$
• $\frac{d}{dx}(-7) = 0$

Answer: $12x^3 - 4x + 5$

Example 2: Find $f'(x)$ for $f(x) = 4e^x - 3\sin x + \ln x$

Apply the specific rules for exponential, trigonometric, and logarithmic functions to each term:

• The derivative of $4e^x$ is $4e^x$.
• The derivative of $-3\sin x$ is $-3\cos x$.
• The derivative of $\ln x$ is $\frac{1}{x}$.

Answer: $f'(x) = 4e^x - 3\cos x + \frac{1}{x}$