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 , and selecting the correct derivative from a set of multiple-choice alternatives.
Given function:
Applying derivative rules:
-
Derivative of :
- Chain rule: , where .
- So, .
-
Derivative of :
- The derivative of is simply .
-
Derivative of 3:
- The derivative of a constant is 0.
Therefore, the derivative of is:
Now, let's check the alternatives from the image to find the matching one:
- Option A: — Incorrect (cosine term should change).
- Option B: — Incorrect (the argument of sine is incorrect).
- Option C: — Incorrect.
- Option D: — Correct.
- Option E: — Incorrect (missing the factor).
Correct answer: Option D.
Do you need more details on the steps?
Here are some related questions to deepen your understanding:
- What is the chain rule, and how is it applied to composite functions like ?
- How do you handle the derivative of an exponential function such as ?
- Can you explain why the derivative of a constant is zero?
- What would be the second derivative of the function ?
- How do you differentiate trigonometric functions involving composite arguments?
Tip: Always check if the function involves composite expressions where the chain rule might apply.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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