Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Ask your questions and receive reliable, detailed answers from our dedicated community of experts.
Sagot :
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]
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]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.