Math Problem Statement

Uma das maneiras de obter a derivada de uma função é pela definição. Porém, dependendo da função, esse processo pode ser bastante longo e complexo. Em contrapartida, existem regras de derivação que facilitam todo esse processo. Com base nas regras de derivação, analise a função a seguir: f(x) = cos(x^2) + e^x + 3. Assinale a alternativa que indica a derivada de 1ª ordem da função f: a) f'(x) = cos(x^2) + e^x; b) f'(x) = -sin(x) + e^x; c) f'(x) = cos(x) + e^x; d) f'(x) = -2x sin(x^2) + e^x; e) f'(x) = -sin(x^2) + e^x

Solution

The question in the image involves calculating the first derivative of the function f(x)=cos(x2)+ex+3f(x) = \cos(x^2) + e^x + 3, and selecting the correct derivative from a set of multiple-choice alternatives.

Given function:

f(x)=cos(x2)+ex+3f(x) = \cos(x^2) + e^x + 3

Applying derivative rules:

  1. Derivative of cos(x2)\cos(x^2):

    • Chain rule: ddx[cos(g(x))]=sin(g(x))g(x)\frac{d}{dx}[\cos(g(x))] = -\sin(g(x)) \cdot g'(x), where g(x)=x2g(x) = x^2.
    • So, ddx[cos(x2)]=sin(x2)2x\frac{d}{dx}[\cos(x^2)] = -\sin(x^2) \cdot 2x.
  2. Derivative of exe^x:

    • The derivative of exe^x is simply exe^x.
  3. Derivative of 3:

    • The derivative of a constant is 0.

Therefore, the derivative of f(x)f(x) is:

f(x)=2xsin(x2)+exf'(x) = -2x \sin(x^2) + e^x

Now, let's check the alternatives from the image to find the matching one:

  • Option A: f(x)=cos(x2)+exf'(x) = \cos(x^2) + e^x — Incorrect (cosine term should change).
  • Option B: f(x)=sin(x)+exf'(x) = -\sin(x) + e^x — Incorrect (the argument of sine is incorrect).
  • Option C: f(x)=cos(x)+exf'(x) = \cos(x) + e^x — Incorrect.
  • Option D: f(x)=2xsin(x2)+exf'(x) = -2x \sin(x^2) + e^x — Correct.
  • Option E: f(x)=sin(x2)+exf'(x) = -\sin(x^2) + e^x — Incorrect (missing the 2x2x factor).

Correct answer: Option D.


Do you need more details on the steps?

Here are some related questions to deepen your understanding:

  1. What is the chain rule, and how is it applied to composite functions like cos(x2)\cos(x^2)?
  2. How do you handle the derivative of an exponential function such as exe^x?
  3. Can you explain why the derivative of a constant is zero?
  4. What would be the second derivative of the function f(x)=cos(x2)+ex+3f(x) = \cos(x^2) + e^x + 3?
  5. How do you differentiate trigonometric functions involving composite arguments?

Tip: Always check if the function involves composite expressions where the chain rule might apply.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Trigonometric Functions
Exponential Functions

Formulas

Derivative of cos(g(x)): -sin(g(x)) * g'(x)
Derivative of e^x: e^x

Theorems

Chain Rule
Basic Differentiation Rules

Suitable Grade Level

Undergraduate Calculus