AP Calculus AB Unit 2 Notes: Learning the Core Differentiation Toolkit (copy)

Power Rule

A rule for differentiating power functions, stating that if $f(x) = x^n$, then $f'(x) = n x^{n-1}$.

Polynomial

An expression composed of variables and coefficients that involves only non-negative integer exponents.

Derivative of a constant

The derivative of any constant is zero, expressed as $\frac{d}{dx}(c) = 0$.

Constant Multiple Rule

The derivative of a constant multiplied by a function is the constant times the derivative of the function: $\frac{d}{dx}(c f(x)) = c f'(x)$.

Sum Rule

The derivative of a sum of functions is the sum of their derivatives: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$.

Difference Rule

The derivative of a difference of functions is the difference of their derivatives: $\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$.

Trigonometric Derivative of Sine

The derivative of the sine function is given by $\frac{d}{dx}(\text{sin}\thickspace x) = \text{cos}\thickspace x$.

Trigonometric Derivative of Cosine

The derivative of the cosine function is given by $\frac{d}{dx}(\text{cos}\thickspace x) = -\text{sin}\thickspace x$.

Exponential Function

A function of the form $a^x$, where the variable is in the exponent.

Base-e Exponential Function

The exponential function where the base is Euler's number e, which has the property that its derivative is itself: $\frac{d}{dx}(e^x) = e^x$.

Natural Logarithm

The logarithm to the base e; its derivative is given by $\frac{d}{dx}(\text{ln} x) = \frac{1}{x}$.

Logarithm of Base a

For base $a$, the derivative of the logarithm function is given by $\frac{d}{dx}(\text{log}_a x) = \frac{1}{x \text{ln}(a)}$.

Negative Exponent Rule

For a function with a negative exponent, such as $f(x) = x^{-n}$, its derivative is $f'(x) = -n x^{-n-1}$.

Fractional Exponent Rule

For a function with a fractional exponent, such as $f(x) = x^{m/n}$, its derivative is $f'(x) = \frac{m}{n} x^{\frac{m}{n} - 1}$.

Units for Logarithmic Functions

The logarithmic functions $\text{ln} x$ and $\text{log}_a x$ are only defined for $x > 0$.

Chain Rule

A differentiation technique used when differentiating a composition of functions.

Tangent Line

A straight line that touches a curve at a point, representing the instantaneous rate of change at that point.

Slope of a Function

The instantaneous rate of change of a function at a given point, represented by its derivative.

Evaluating Derivatives at a Point

To determine the slope of a function at a specific value $x=a$ by computing $f'(a)$.

Common Mistake: Forgetting Negative Sign

A frequent error in differentiating $\text{cos} x$, leading to writing $\frac{d}{dx}(\text{cos} x) = \text{sin} x$ instead of the correct $-\text{sin} x$.