MH1810 Mathematics 1 - Chapter 5 Differentiation Flashcards

Vocabulary flashcards covering differentiation rules, limits, derivatives of trigonometric and inverse trigonometric functions, parametric and implicit differentiation, logarithmic differentiation, linearization, differentials, and rates of change.

Tangent line

A straight line passing through a point $P$ on a curve that touches the curve at $P$ and meets it at only one point in a sufficiently small neighborhood around $P$.

Secant line (Chord)

A straight line passing through any two points on a curve.

Increment

The change in a variable $x$, represented by $\Delta x = \text{new value of } x - \text{old value of } x$.

Average rate of change

The ratio $\frac{\Delta f}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$ representing the change in $f$ per unit change in $x$ over an interval.

Instantaneous rate of change

The limit $\lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}$, representing the rate of change of $f$ with respect to $x$ at a specific moment.

Derivative at a point

The limit $f'(c) = \lim_{x \rightarrow c} \frac{f(x) - f(c)}{x - c}$, provided this limit exists.

Differentiable function on an open interval

A function $f$ that is differentiable at every number $c$ in the open interval $(a, b)$.

Differentiability and Continuity Theorem

A theorem stating that if a function $f$ is differentiable at $x = c$, then $f$ is continuous at $x = c$.

Sum Rule for Differentiation

The rule stating that if $f$ and $g$ are differentiable at $x = c$, then $(f + g)'(c) = f'(c) + g'(c)$.

Difference Rule for Differentiation

The rule stating that if $f$ and $g$ are differentiable at $x = c$, then $(f - g)'(c) = f'(c) - g'(c)$.

Product Rule for Differentiation

The rule stating that if $f$ and $g$ are differentiable at $x = c$, then $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$.

Quotient Rule for Differentiation

The rule stating that if $f$ and $g$ are differentiable at $x = c$ and $g(c) \neq 0$, then $\left(\frac{f}{g}\right)'(c) = \frac{f'(c)g(c) - f(c)g'(c)}{(g(c))^2}$.

Reciprocal Rule for Differentiation

The rule stating that if $g$ is differentiable at $x = c$ and $g(c) \neq 0$, then $\left(\frac{1}{g}\right)'(c) = -\frac{g'(c)}{(g(c))^2}$.

Chain Rule

The rule stating that if $g(x)$ is differentiable at $x = c$ and $f(u)$ is differentiable at $u = g(c)$, then $(f \circ g)'(c) = f'(g(c))g'(c)$, or in Leibniz notation, $\frac{df}{dx} = \frac{df}{du} \times \frac{du}{dx}$.

Derivative of $\sin(x)$

$$\frac{d}{dx}(\sin(x)) = \cos(x)$$

Derivative of $\cos(x)$

$$\frac{d}{dx}(\cos(x)) = -\sin(x)$$

Derivative of $\tan(x)$

$$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

Derivative of $\sec(x)$

$$\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$$

Derivative of $\csc(x)$

$$\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$$

Derivative of $\cot(x)$

$$\frac{d}{dx}(\cot(x)) = -\csc^2(x)$$

Derivative of $e^x$

$\frac{d}{dx}(e^x) = e^x$, making $e^x$ the only function that is equal to its own derivative.

Derivative of $a^x$

$\frac{d}{dx}(a^x) = a^x \ln(a)$ for $a > 0$ and $a \neq 1$.

Derivative of $\ln(x)$

$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$ for $x > 0$.

Derivative of $\log_a(x)$

$\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$ for $a > 0$ and $a \neq 1$.

Parametric Differentiation

A technique to differentiate variables $x = v(t)$ and $y = u(t)$ with respect to each other using $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{u'(t)}{v'(t)}$, provided $v'(t) \neq 0$.

Implicit Differentiation

A process using the chain rule to find $\frac{dy}{dx}$ when $x$ and $y$ are related by an equation $F(x, y) = 0$ without solving for $y$ explicitly.

Power Rule for Rational Exponents

The theorem stating that for $r = \frac{m}{n}$ (where $n \in \mathbb{Z}^+$ and $m \in \mathbb{Z}$), $\frac{d}{dx}(x^r) = r x^{r-1}$.

Logarithmic Differentiation

A technique where the natural logarithm $\ln(x)$ is taken on both sides of an equation $y = f(x)$ before performing implicit differentiation, useful for functions with variable bases and exponents such as $y = x^x$.

Derivative of Inverse Function Theorem

A theorem stating that if $f$ is continuous and strictly monotonic with $f'(x_0) \neq 0$, then $(f^{-1})'(y_0) = \frac{1}{f'(f^{-1}(y_0))} = \frac{1}{f'(x_0)}$ where $y_0 = f(x_0)$.

Derivative of $\sin^{-1}(x)$

$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}}$ for $-1 < x < 1$.

Derivative of $\cos^{-1}(x)$

$\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1 - x^2}}$ for $-1 < x < 1$.

Derivative of $\tan^{-1}(x)$

$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1 + x^2}$ for $x \in \mathbb{R}$.

Derivative of $\cot^{-1}(x)$

$\frac{d}{dx}(\cot^{-1}(x)) = -\frac{1}{1 + x^2}$ for $x \in \mathbb{R}$.

Derivative of $\sec^{-1}(x)$

$\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{|x|\sqrt{x^2 - 1}}$ for $x < -1$ or $x > 1$.

Derivative of $\csc^{-1}(x)$

$\frac{d}{dx}(\csc^{-1}(x)) = -\frac{1}{|x|\sqrt{x^2 - 1}}$ for $x < -1$ or $x > 1$.

Linearization of $f$ at $a$

The linear function $L(x) = f(a) + f'(a)(x - a)$ used as a local linear approximation for $f(x)$ near $x = a$.

Differential of $f$ ($df$)

The estimated change in $f$ given by $df = L(a + dx) - L(a) = f'(a)dx$ for a small change $dx$.

Relative Change

The ratio of change to the initial value, where actual relative change is $\frac{\Delta f}{f(a)}$ and estimated relative change is $\frac{df}{f(a)}$.

Percentage Change

The relative change expressed as a percentage, where actual percentage change is $100 \times \frac{\Delta f}{f(a)}$ and estimated percentage change is $100 \times \frac{df}{f(a)}$.

Higher Derivatives

Derivatives obtained by repeatedly differentiating a function, such as the second derivative $f''(x) = \frac{d^2y}{dx^2}$, third derivative $f'''(x) = \frac{d^3y}{dx^3}$, and so on.