To differentiate the expression [tex]\(\sin(x) - x^3 \cos(x)\)[/tex] with respect to [tex]\(x\)[/tex], we will need to use the differentiation rules for trigonometric functions and the product rule. Here is the step-by-step differentiation process:
Given:
[tex]\[ f(x) = \sin(x) - x^3 \cos(x) \][/tex]
Step 1: Differentiate [tex]\(\sin(x)\)[/tex].
The derivative of [tex]\(\sin(x)\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[ \frac{d}{dx}[\sin(x)] = \cos(x) \][/tex]
Step 2: Differentiate [tex]\(-x^3 \cos(x)\)[/tex] using the product rule.
The product rule states that the derivative of a product [tex]\(u(x) v(x)\)[/tex] is [tex]\(u'(x)v(x) + u(x)v'(x)\)[/tex], where [tex]\(u(x) = -x^3\)[/tex] and [tex]\(v(x) = \cos(x)\)[/tex].
First, differentiate [tex]\(-x^3\)[/tex]:
[tex]\[ \frac{d}{dx}[-x^3] = -3x^2 \][/tex]
Next, differentiate [tex]\(\cos(x)\)[/tex]:
[tex]\[ \frac{d}{dx}[\cos(x)] = -\sin(x) \][/tex]
Applying the product rule, we get:
[tex]\[ \frac{d}{dx}[-x^3 \cos(x)] = (-3x^2) \cos(x) + (-x^3) (-\sin(x)) \][/tex]
[tex]\[ = -3x^2 \cos(x) + x^3 \sin(x) \][/tex]
Step 3: Combine the results.
Adding the derivatives from Steps 1 and 2:
[tex]\[ f'(x) = \cos(x) + (-3x^2 \cos(x) + x^3 \sin(x)) \][/tex]
[tex]\[ f'(x) = \cos(x) - 3x^2 \cos(x) + x^3 \sin(x) \][/tex]
Step 4: Simplify the expression.
Combine like terms:
[tex]\[ f'(x) = x^3 \sin(x) - 3 x^2 \cos(x) + \cos(x) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{x^3 \sin(x) - 3 x^2 \cos(x) + \cos(x)} \][/tex]
This matches with option [tex]\(b\)[/tex]:
[tex]\[ \boxed{(1-3 x) \cos x+x^3 \sin x} \][/tex]
