Math / Calculus

QuestionGiven the function $f(x)=5 x^{2} \cos x$ , find $f^{\prime}(x)$ in any form.
Answer Attempt 1 out of 2 $f^{\prime}(x)=$ $\square$ Submit Answer

Studdy Solution

STEP 1

1. We are given the function $f(x) = 5x^2 \cos x$ .
2. We need to find the derivative $f'(x)$ .

STEP 2

1. Identify the rule for differentiation to be used.
2. Apply the product rule to differentiate $f(x)$ .
3. Simplify the expression for $f'(x)$ .

STEP 3

Identify the rule for differentiation: We will use the product rule, which states that if $u(x)$ and $v(x)$ are functions of $x$ , then the derivative of their product is given by:
$(uv)' = u'v + uv'$
In this case, let $u(x) = 5x^2$ and $v(x) = \cos x$ .

STEP 4

Differentiate $u(x) = 5x^2$ :
$u'(x) = \frac{d}{dx}(5x^2) = 10x$

STEP 5

Differentiate $v(x) = \cos x$ :
$v'(x) = \frac{d}{dx}(\cos x) = -\sin x$

STEP 6

Apply the product rule:
$f'(x) = u'(x)v(x) + u(x)v'(x)$ $f'(x) = (10x)(\cos x) + (5x^2)(-\sin x)$

STEP 7

Simplify the expression for $f'(x)$ :
$f'(x) = 10x \cos x - 5x^2 \sin x$
The derivative of the function $f(x) = 5x^2 \cos x$ is:
$f'(x) = 10x \cos x - 5x^2 \sin x$

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QuestionGiven the function $f(x)=5 x^{2} \cos x$ , find $f^{\prime}(x)$ in any form.
Answer Attempt 1 out of 2 $f^{\prime}(x)=$ $\square$ Submit Answer

Studdy Solution

STEP 1

1. We are given the function $f(x) = 5x^2 \cos x$ .
2. We need to find the derivative $f'(x)$ .

STEP 2

1. Identify the rule for differentiation to be used.
2. Apply the product rule to differentiate $f(x)$ .
3. Simplify the expression for $f'(x)$ .

STEP 3

Identify the rule for differentiation: We will use the product rule, which states that if $u(x)$ and $v(x)$ are functions of $x$ , then the derivative of their product is given by:
$(uv)' = u'v + uv'$
In this case, let $u(x) = 5x^2$ and $v(x) = \cos x$ .

STEP 4

Differentiate $u(x) = 5x^2$ :
$u'(x) = \frac{d}{dx}(5x^2) = 10x$

STEP 5

Differentiate $v(x) = \cos x$ :
$v'(x) = \frac{d}{dx}(\cos x) = -\sin x$

STEP 6

Apply the product rule:
$f'(x) = u'(x)v(x) + u(x)v'(x)$ $f'(x) = (10x)(\cos x) + (5x^2)(-\sin x)$

STEP 7

Simplify the expression for $f'(x)$ :
$f'(x) = 10x \cos x - 5x^2 \sin x$
The derivative of the function $f(x) = 5x^2 \cos x$ is:
$f'(x) = 10x \cos x - 5x^2 \sin x$

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QuestionGiven the function f(x)=5x2cos⁡xf(x)=5 x^{2} \cos xf(x)=5x2cosx, find f′(x)f^{\prime}(x)f′(x) in any form.Answer Attempt 1 out of 2 f′(x)=f^{\prime}(x)=f′(x)= □\square□ Submit Answer

Studdy Solution

STEP 1

1. We are given the function f(x)=5x2cos⁡x f(x) = 5x^2 \cos x f(x)=5x2cosx.2. We need to find the derivative f′(x) f'(x) f′(x).

STEP 2

1. Identify the rule for differentiation to be used.2. Apply the product rule to differentiate f(x) f(x) f(x).3. Simplify the expression for f′(x) f'(x) f′(x).

STEP 3

STEP 4

Differentiate u(x)=5x2 u(x) = 5x^2 u(x)=5x2:u′(x)=ddx(5x2)=10x u'(x) = \frac{d}{dx}(5x^2) = 10x u′(x)=dxd​(5x2)=10x

STEP 5

Differentiate v(x)=cos⁡x v(x) = \cos x v(x)=cosx:v′(x)=ddx(cos⁡x)=−sin⁡x v'(x) = \frac{d}{dx}(\cos x) = -\sin x v′(x)=dxd​(cosx)=−sinx

STEP 6

Apply the product rule:f′(x)=u′(x)v(x)+u(x)v′(x) f'(x) = u'(x)v(x) + u(x)v'(x) f′(x)=u′(x)v(x)+u(x)v′(x) f′(x)=(10x)(cos⁡x)+(5x2)(−sin⁡x) f'(x) = (10x)(\cos x) + (5x^2)(-\sin x) f′(x)=(10x)(cosx)+(5x2)(−sinx)

STEP 7

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QuestionGiven the function $f(x)=5 x^{2} \cos x$ , find $f^{\prime}(x)$ in any form.
Answer Attempt 1 out of 2 $f^{\prime}(x)=$ $\square$ Submit Answer

1. We are given the function $f(x) = 5x^2 \cos x$ .
2. We need to find the derivative $f'(x)$ .

1. Identify the rule for differentiation to be used.
2. Apply the product rule to differentiate $f(x)$ .
3. Simplify the expression for $f'(x)$ .

Differentiate $u(x) = 5x^2$ :
$u'(x) = \frac{d}{dx}(5x^2) = 10x$

Differentiate $v(x) = \cos x$ :
$v'(x) = \frac{d}{dx}(\cos x) = -\sin x$

Apply the product rule:
$f'(x) = u'(x)v(x) + u(x)v'(x)$ $f'(x) = (10x)(\cos x) + (5x^2)(-\sin x)$