Respuesta :
The derivatives for this problem are given as follows:
a) [tex]f^{\prime}(x) = -4\sin{x} + \frac{1}{x + 1}[/tex]
b) [tex]f^{\prime}(x) = \sec{x}\tan^{2}{x} + \sec^3{x}[/tex].
What is the derivative of the sum?
The derivative of the sum is the sum of the derivatives.
In this problem, the function is:
[tex]f(x) = 4\cos{x} + \ln{(x + 1)}[/tex]
Using a derivative table for the derivatives of the cosine and the ln, the derivative of the function is:
[tex]f^{\prime}(x) = (4\cos{x})^{\prime} + (\ln{(x + 1)})^{\prime}[/tex]
[tex]f^{\prime}(x) = -4\sin{x} + \frac{1}{x + 1}[/tex]
What is the product rule?
The derivative of the product is given as follows:
[tex](f(x) \times g(x))^{\prime} = f^{\prime}(x)g(x) + g^{\prime}(x)f(x)[/tex]
In this problem, we have that:
- [tex]f(x) = \sec{x}, f^{\prime}(x) = \sec{x}\tan{x}[/tex].
- [tex]g(x) = \tan{x}, f^{\prime}(x) = \sec^2{x}[/tex].
Hence the derivative is:
[tex]f^{\prime}(x) = \sec{x}\tan^{2}{x} + \sec^3{x}[/tex].
More can be learned about derivatives at https://brainly.com/question/2256078
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