Unit 2: Differentiation: Definition and Basic Rules

15 terms. Say what each one means, then open the card to check.

• Average rate of change

  The change in output over the change in input on an interval, $\frac{f(b) - f(a)}{b - a}$. It is the slope of the secant line through the two endpoints.

  Example. A hiker who climbs from 1,200 meters at 9 a.m. to 1,500 meters at 11 a.m. gains height at an average rate of 150 meters per hour.

• Derivative at a point

  The limit $f'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}$, when it exists. It is the instantaneous rate of change of $f$ at $a$.

  Example. For $f(x) = 5x$, every difference quotient is $\frac{5(a + h) - 5a}{h} = 5$, so $f'(a) = 5$ at every $a$.

• Tangent line

  The line through $(a, f(a))$ with slope $f'(a)$: $y = f(a) + f'(a)(x - a)$. Near $a$ it is the line that best matches the graph.

  Example. If $f(4) = 10$ and $f'(4) = -2$, the tangent line at $x = 4$ is $y = 10 - 2(x - 4)$.

• Derivative notation

  The derivative of $y = f(x)$ can be written $f'(x)$, $y'$, $\frac{dy}{dx}$ or $\frac{d}{dx}f(x)$. All mean the same function.

  Example. If $y = x^2$, then $y' = 2x$, $\frac{dy}{dx} = 2x$ and $\frac{d}{dx}(x^2) = 2x$ say the same thing.

• Differentiable

  A function is differentiable at $a$ when $f'(a)$ exists. If $f$ is differentiable at $a$, it is continuous at $a$; a continuous function need not be differentiable.

  Example. $|x|$ is continuous at 0, but its slopes are -1 on the left and 1 on the right, so it is not differentiable at 0.

• Where a derivative fails

  A function is not differentiable at a corner (one-sided slopes differ), at a cusp, at a vertical tangent line, or anywhere it is not continuous.

  Example. $\sqrt[3]{x}$ has a vertical tangent at $x = 0$: its graph is smooth there but the slope grows without bound, so no derivative exists at 0.

• Power rule

  For any real number $n$, $\frac{d}{dx}x^n = nx^{n-1}$. Rewrite roots and reciprocals as powers first.

  Example. $\frac{1}{x^3} = x^{-3}$, so its derivative is $-3x^{-4} = -\frac{3}{x^4}$.

• Sum and constant multiple rules

  Differentiate a sum term by term, keep a constant factor, and drop a constant term: $(cf + g)' = cf' + g'$ and $\frac{d}{dx}c = 0$.

  Example. $\frac{d}{dx}(7x^2 - 4x + 11) = 14x - 4$; the 11 contributes nothing.

• Derivatives of $e^x$ and $\ln x$

  $\frac{d}{dx}e^x = e^x$, and $\frac{d}{dx}\ln x = \frac{1}{x}$ for $x > 0$.

  Example. $\frac{d}{dx}(2e^x + \ln x) = 2e^x + \frac{1}{x}$, which is $2e + 1$ at $x = 1$.

• Product rule

  $(fg)' = f'g + fg'$: the derivative of the first times the second, plus the first times the derivative of the second.

  Example. $\frac{d}{dx}(x^2\ln x) = 2x\ln x + x^2 \cdot \frac{1}{x} = 2x\ln x + x$.

• Quotient rule

  $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ wherever $g \ne 0$. The order on top matters, because of the minus sign.

  Example. $\frac{d}{dx}\frac{x}{x + 1} = \frac{1 \cdot (x + 1) - x \cdot 1}{(x + 1)^2} = \frac{1}{(x + 1)^2}$.

• Derivatives of sine and cosine

  With $x$ in radians, $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$.

  Example. The slope of $y = \sin x$ at $x = 0$ is $\cos 0 = 1$, and at $x = \frac{\pi}{2}$ it is $\cos\frac{\pi}{2} = 0$, the top of the wave.

• Derivatives of tangent and secant

  $\frac{d}{dx}\tan x = \sec^2 x$ and $\frac{d}{dx}\sec x = \sec x\tan x$. Both follow from the quotient rule.

  Example. At $x = 0$, the slope of $y = \tan x$ is $\sec^2 0 = 1$.

• Derivatives of cotangent and cosecant

  $\frac{d}{dx}\cot x = -\csc^2 x$ and $\frac{d}{dx}\csc x = -\csc x\cot x$. The co- functions take a minus sign.

  Example. At $x = \frac{\pi}{2}$, the slope of $y = \cot x$ is $-\csc^2\frac{\pi}{2} = -1$.

• Estimating a derivative from a table

  Use the two table inputs closest to $a$, one on each side, and compute the slope between them. It is an estimate, not the exact derivative.

  Example. If $W(10) = 48$ and $W(14) = 60$, then $W'(12) \approx \frac{60 - 48}{14 - 10} = 3$.