2.1: The Derivative and Tangent Line Problem

The Tangent Line Problem

The problem of finding the tangent line at a point P boils down to the problem of finding the slope of the tangent line at point P. You can approximate this slope using a secant line through the point of tangency and a second point on the curve.

If (c,\(f\)(c)) is the point of tangency and \( ((c)+Δx, f(c+Δx)) \) is a second point on the graph of \(f\), then the slope of the secant line through the two points is given by substitution into the slope formula.

\(m = \frac{y_2-y_1}{x_2-x_1}\)

\(m_{sec}= \frac{f(c+Δx)-f(c)}{(c+Δx)-c}\) \(\frac{Change\space in\space y}{Change\space in\space x}\)

\(m_{sec}=\frac{f(c + Δx) - f(c)}{Δx}.\) \(slope\space of\space secant\space line\)

Definition of Tangent Line with Slope m

If \(f\) is defined on an open interval containing c, and if the limit

$$\lim_{Δx\to0}\frac{Δy}{Δx} = \lim_{Δx\to0}\frac{f(c + Δx) - f(c)}{Δx}=m$$

exists, then the line passing through (c,\(f\)(c)) with slope m is the tangent line to the graph of \(f\) at the point c.

Definition of a Vertical Tangent Line

If \(f\) is continuous at c and

$$\lim_{Δx\to0}\frac{f(c + Δx) - f(c)}{Δx}=∞ \space\space\space\space or \space\space\space\space \lim_{Δx\to0}\frac{f(c + Δx) - f(c)}{Δx}=-∞,$$

then the vertical line x = c passing through (c, \(f\)(c)) is a vertical tangent line to the graph of \(f\).

The Derivative of a Function

Definition of the Derivative of a Function

The derivative of \(f\) at x is

$$f'(x) = \lim_{Δx\to0}\frac{f(x + Δx) - f(x)}{Δx}$$

provided the limit exists. For all x for which this limit exists, \(f\)' is a functon of x.

The process of finding the derivative of a function is called differentiation. A function is differentiable when its derivative exists at x and is differentiable on an open interval when it is differentiable on any point of the interval.

Note: The equation of a tangent line at any given point is found by taking the function's derivative. Similarly, the slope of a tangent line at point (c,\(f\)(c)) is found by evaluating \(f\)'(c).

Notation

The most common notations for denoting the derivative of \(f\)(x) = y are:

$$ f'(x), \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{dy}{dx}, \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space y', \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space D_x[y] $$

Using limit notation, you can write:

$$\frac{dy}{dx}=\lim_{Δx\to0}\frac{Δy}{Δx}=\lim_{Δx\to0}\frac{f(x + Δx) - f(x)}{Δx}=f'(x)$$

Differentiability and Continuity

The alternative limit form of the derivative is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is

$$f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$$

provided this limit exists.

The existence of the limit in this alternative form requires that the one-sided limits exist and are equal. These one-sided limits are called the derivatives from the left and from the right, respectively. It follows that f is differentiable on the closed interval [a,b] when it is differentiable on (a,b) and when the derivative from the right at a and the derivative from the left at b both exist.

Differentaiblity Implies Continuity

If \(f\) is differentiable at x = c, then \(f\) is continuous at x = c.

If \(f\) is not continuous at x = c, then \(f\) is not differentiable at x = c.

Note: Only these affirmations are always true.

Extra Notes:

2.2: Basic Differentation Rules and Rates of Change

Basic Differentiation Rules

The Constant Rule

The derivative of a constant function is 0. That is, if c is a real number.

$$\frac{d}{dx}[c] = 0$$

The Power Rule

General Binomial Expansion

The general binomial expansion for a positive integer n is

$$(x+Δx)^n = x^n+nx^{n-1}+\frac{n(n-1)x^{n-2}}{2}(Δx)^2+\dotsb+(Δx)^n$$

Power Rule

If n is a rational number, then the function \( f(x) + x^n \) is differentaible and

$$(x+Δx)^n = x^n+nx^{n-1}+\frac{n(n-1)x^{n-2}}{2}(Δx)^2+\dotsb+(Δx)^n$$

The Constant Multiple Rule

If \(f\) is a differentiable function and c is a real number, then c\(f\) is also differentiable and

$$\frac{d}{dx}[cf(x)]=cf'(x)$$

The Sum and Difference Rules

The sum of two differentiable functions \(f\) and g is itself differentiable. Moreover, the derivative of \(f\) + \(g\) is the sum of the derivatives of \(f\) and \(g\). Same thing applies to differences.

$$\frac{d}{dx}[f(x)±g(x)]=f'(x)±g'(x)$$

Derivatives of Sine and Cosine Functions

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

Derivative of the Natural Exponential Function

$$\frac{d}{dx}[e^x]=e$$

Rates of Change (Instantaneous Velocity)

If x then x(t)

If y then y(t)

Note: the following MUST be shown on the exam:

s'(t) = v(t)

s''(t) = v'(t) = a(t)

Extra Notes:

2.3: Product and Quotient Rules and High-Order Derivatives

Product and Quotient Rules

The Product Rule

The product of two differentiable functions \(f\) and \(g\) is itself differentiable. Moreover, the derivative of \(f\)\(g\) is the first function times the derivative of the second, plus the second function times the derivative of the first.

$$\frac{d}{dx}[f(x)g'(x) + g(x)f'(x)] = f(x)g'(x) + g(x)f'(x)$$

The Quotient Rule

The quotient \(f\)/\(g\) of two differentiable functions \(f\) and \(g\) is itself differentiable at all values of x for which \(g\)(x) ≠ 0. Moreover, the derivative of \(f\)/\(g\) is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

$$\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2},\space g(x)≠0$$

Derivatives of Trigonometric Functions

$$ \frac{d}{dx}[\tan(x)]= \sec^2(x) \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[\cot(x)]=-\csc^2(x) $$$$ \frac{d}{dx}[\sec(x)]= \sec(x)\tan(x) \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[\csc(x)]=-\csc(x)\cot(x) $$

Higher-Order Derivatives

Note (again): You MUST (unless the problem states otherwise) declare velocity as s'(x) = v(x) and acceleration as s''(x) = v'(x) = a(x).

Notation

First Derivative: \( \space\space\space\space\space\space\space y', f'(x), \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{dy}{dx}, \space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space\space\space D_x[y]\)

Second derivative:\( \space\space\space\space y'', f''(x), \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{d^2y}{dx^2}, \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^2}{dx^2}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space D_x^2[y]\)

Third derivative:\( \space\space\space\space\space\space\space y''', f'''(x), \space\space\space\space\space\space\space\space\space\space\space\space\space\space \frac{d^3y}{dx^3}, \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^3}{dx^3}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space D_x^3[y]\)

Fourth derivative:\( \space\space\space\space y^{(4)}, f^{(4)}(x), \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^4y}{dx^4}, \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^4}{dx^4}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space D_x^4[y]\)

nth derivative:\( \space\space\space\space\space\space\space\space y^{(n)}, f^{(n)}(x), \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^ny}{dx^n}, \space\space\space\space\space\space\space\space\space\space\space\space \frac{d^n}{dx^n}[f(x)], \space\space\space\space\space\space\space\space\space\space\space\space D_x^n[y]\)

Extra Notes:

Super simple just remember to remember signs and the squared function at the bottom for the quotient rule.

2.4: The Chain Rule

The Chain Rule

If y = \(f\)(u) is a differentiable funcition of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and

$$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

or, equivalently,

$$\frac{d}{dx}[f(g(x))]=f'(g(x))g'(x)$$

The General Power Rule

If y = (u(x))^n, where u is a differentiable function of x and n is a rational number, then

$$\frac{d}{dx}[u^n]=nu^{n-1}u'$$

Derivative of a Logarithmic Function

Let u be a differentiable function of x.

\( \frac{d}{dx}[\ln x] = \frac{1}{x},\space x>0 \)

$$\frac{d}{dx}[\ln u] = \frac{1}{u}\frac{du}{dx}=\frac{u'}{u},\space x > 0$$

Derivatives for Bases Other than e

$$\frac{d}{dx}[a^x]=(\ln a)a^x$$ $$\frac{d}{dx}[a^u]=(\ln a)a^u\cdot\frac{du}{dx}$$ $$\frac{d}{dx}[\log_a x]=\frac{1}{(\ln a)x}$$ $$\frac{d}{dx}[\log_a u]=\frac{1}{(\ln a)u}\cdot\frac{du}{dx}$$

Extra Notes:

Never confuse variables for constants...

2.5 Implicit Differentiation

Implicit Differentiation is used to differentiate any kind of relation between anything. Its commonly used to differentiate relations for x and y that aren't functions or aren't very straightforward.

Guidelines for Implicit Differentiations

1. Differentiate both sides of the equation with respect to x.
2. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.
3. Factor dy/dx out of the left side of the equation.
4. Solve for dy/dx.

Notation for finding implicit derivatives at a point:

$$\frac{dy}{dx}|_{x_1,y_1}=$$

Extra Notes:

2.6 Derivatives of an Inverse Function

Continuity and Differentiability of Inverse Functions

Let \(f\) be a function that is differentiable on some interval i. If \(f\) has an inverse function, then the following statements are true.

1. If \(f\) is continuous on its domain, then \(f\)$^{-1}$ is continuous on its domain.
2. 2. If \(f\) is differentiable on an interval containing c and \(f\)'(c) ≠ 0, then \(f\)$^{-1}$ is differentiable at \(f\)'(c).

The Derivative of an Inverse Function

Let \(f\) be a function that is differentiable on some interval i. If \(f\) has an inverse function \(g\), then \(g\) is differentiable at any x for which \(f\)'(\(g\)(x)) ≠ 0. Moreover,

$$g'(x)=\frac{1}{f'(g(x))},\space f'(g(x))≠0$$

Graphs of Inverse Functions Have Reciprocal Slopes

If \(f\)(a) = b, then \(f^{-1}\)(b) = a.

Using the rule seen above, we can conclude that \(g'(a)=\frac{1}{f'(b)}\).

Derivatives of Inverse Trigonometric Functions

Let u be a differentiable function of x.

\( \frac{d}{dx}[\sin^{-1}u]=\frac{u'}{\sqrt{1-u^2}} \space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[\cos^{-1}u]=\frac{-u'}{\sqrt{1-u^2}} \)

\( \frac{d}{dx}[\tan^{-1}u]= \frac{u'}{1+u^2} \space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[\cot^{-1}u]= \frac{-u'}{1+u^2} \)

\( \frac{d}{dx}[\sec^{-1}u]=\frac{u'}{|u|\sqrt{u^2-1}} \space\space\space\space\space\space\space\space\space\space\space\space \frac{d}{dx}[\csc^{-1}u]=\frac{-u'}{|u|\sqrt{u^2-1}} \)

Extra Notes:

Its not necessary to find the inverse function first. If you have $f$(x) equal to a constant b, then you can simply solve for x to find \( f^{-1}$(b) \).

2.7: Related Rates

An important use of the Chain Rule is to find the rates of change.

Guidelines for solving Related-Rate Problems

1. Identify all given quantities and quantities to be determined. Make a setch and label the quatities.
2. Write an equation involving the variables whose rates of change either are given or are to be determined.
3. Using the Chain Rule, implicity differentiate both sides of the equation with respect to time t.
4. After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change.

Extra Notes:

Related Rates is just "Problemas Razonados" using everything in the past topics.

Its super useful to know the formuli of areas and volumes of different polygons and polyhedra.

Its incredibly useful to remember old concepts like 'semejanza', trigonometric identities, and, of course, pythagoras.