Derive y = ln(sent)

Passo 1

Beleza, vamos resolver esse exercício onde precisamos derivar a função . Aqui, estamos trabalhando com cálculo diferencial, especificamente usando as regras de derivação. Para resolver isso, vamos precisar conhecer a regra da cadeia e saber como derivar funções logarítmicas e trigonométricas.

Passo 2

Primeiro, vamos identificar a função que queremos derivar: . Essa é uma função composta, porque dentro do logaritmo natural temos uma função trigonométrica, . Quando temos funções compostas, usamos a regra da cadeia para derivar.

Passo 3

A regra da cadeia nos diz que, se temos uma função composta , a derivada de em relação a é:
No nosso caso, e . Vamos encontrar as derivadas de cada uma delas.

Passo 4

Vamos derivar a função externa :
Agora, derivamos a função interna :

Passo 5

Aplicando a regra da cadeia, substituímos e na fórmula:

Passo 6

Simplificando a expressão, temos:
E sabemos que é igual a (cotangente de t).

Resposta

A derivada da função é:

Exercícios relacionados

Calcule a integral dada, colocando-a em coordenadas polares.
, onde R é a região do primeiro quadrante limitada pelo círculo x^2 + y^2 = 25
Utilize coordenadas polares para de
<html><head></head><body><div>Esboce o gráfico de <span class="math-inline ">
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<html><head></head><body><div>Dtermine o retângulo de área máxima e lados paralelos aos eixos coordenados inscrito ua clipse <span class="math-inline ">
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