Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.5.54

Capitolo 3, Problema 3.5.54

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x

Guida verificata passo dopo passo

Start by recalling the definition of the cosecant function: \( \csc x = \frac{1}{\sin x} \). We need to find the derivative of this function with respect to \( x \).

Apply the Quotient Rule for derivatives, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = \sin x \).

Calculate the derivatives: \( u' = 0 \) since the derivative of a constant is zero, and \( v' = \cos x \) since the derivative of \( \sin x \) is \( \cos x \).

Substitute these into the Quotient Rule formula: \( \frac{\sin x \cdot 0 - 1 \cdot \cos x}{(\sin x)^2} = \frac{-\cos x}{\sin^2 x} \).

Recognize that \( \frac{-\cos x}{\sin^2 x} \) can be rewritten using trigonometric identities as \( -\csc x \cdot \cot x \), thus verifying the derivative formula \( \frac{d}{dx}(\csc x) = -\csc x \cot x \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Quotient Rule

The Quotient Rule is a fundamental technique in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be calculated using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential for verifying derivatives of functions like csc(x), which can be expressed as a quotient.

Video consigliato:

06:43

The Quotient Rule

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding the properties and behavior of the cosecant function is crucial for differentiating it, as it influences the application of the Quotient Rule. Additionally, knowing the relationship between csc(x) and sin(x) helps in simplifying the derivative calculations.

Video consigliato:

Percorso guidato

6:22

Graphs of Secant and Cosecant Functions

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. Key identities, such as sin^2(x) + cos^2(x) = 1, are often used in calculus to simplify expressions and derivatives. Familiarity with these identities is important when verifying derivative formulas, as they can help in transforming and simplifying the resulting expressions from the differentiation process.

Video consigliato:

7:17

Verifying Trig Equations as Identities

Pratica correlata

Domanda del libro di testo

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

187

views

Domanda del libro di testo

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

349

views

Domanda del libro di testo

Find an equation of the line tangent to the curve y = sin x at x = 0.

352

views