Calculus Notes
⟵Section 2.1 Section 2.3 ⟶Section 2.2 Basic Differentiation Rules and Rates of Change
This section introduces several important shortcuts that can be used to calculate derivatives without resorting to the definition.
The constant rule states that if $c$ is constant, then
$$\boxed{\frac{\diff}{\diff x} c = 0}$$"The derivative of a constant is zero"
The power rule states that
$$\boxed{\frac{\diff}{\diff x} x^n = nx^{n-1}}$$"Bring the exponent down to the front and reduce by one."
The constant multiple rule states that if $c$ is constant, then
$$\boxed{\frac{\diff}{\diff x} \left[ c f(x)\right] = c \frac{\diff}{\diff x} f(x)}$$"Constants may be factored out of the derivative."
The sum and difference rules states that
$$\boxed{\frac{\diff}{\diff x} \left [f(x) \pm g(x) \right] = f'(x) \pm g'(x)}$$"The derivative of a sum is the sum of the derivatives."
The derivatives of $\sin x$ and $\cos x$ are
$$\boxed{\frac{\diff}{\diff x} \sin x = \cos x} \hspace{20pt} \boxed{\frac{\diff}{\diff x} \cos x = -\sin x}$$"The derivative of sine is cosine, and vice versa; the derivative of any co-function is negative."
Rates of change
It is natural to think of a derivative as a slope, but a more general interpretation of the derivative is therate of change of one variable with respect to another. In other words, $\frac{\diff f}{diff x}$ means "how fast $f$ changes when $x$ is varied". In particular, if $s$ represents displacement and $t$ represents time, then the derivative $\frac{\diff s}{\diff t}$ represents velocity. (As you may have learned in physics, velocity an be either positive or negative, but speed is the absolute value of velocity.)
The exact equations governing the relationship between displacement and velocity depend on the displacement function. Of particular importance is the case where acceleration $a$ is constant. In this case:
$$s(t) = \frac{1}{2} a t^2 + v_0 t + s_0$$where $v_0$ and $s_0$ are the initial velocity and displacement, respectively. In the case of a free-falling object $a = g \approx -9.8 \text{m/s}^2$.
⟵Section 2.1 Section 2.3 ⟶