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Math  /  Calculus

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Given the function h(x)=−3x6x3−3h(x)=-3 x \sqrt{6 x^{3}-3}, find h′(x)h^{\prime}(x) in any form.

Studdy Solution

STEP 1

1. The function h(x)=−3x6x3−3 h(x) = -3x \sqrt{6x^3 - 3} involves a product and a composite function.
2. To find h′(x) h'(x) , we need to use the product rule and the chain rule of differentiation.
3. We will differentiate each part of the product separately and then combine the results using the product rule.

STEP 2

1. Identify and label the parts of the function suitable for the product rule.
2. Differentiate each part of the product separately.
3. Apply the product rule to combine the results.
4. Simplify the final expression for h′(x) h'(x) .

STEP 3

Identify the parts of the function h(x)=−3x6x3−3 h(x) = -3x \sqrt{6x^3 - 3} .
Let u(x)=−3x u(x) = -3x and v(x)=6x3−3 v(x) = \sqrt{6x^3 - 3} .

STEP 4

Differentiate u(x) u(x) with respect to x x .
u(x)=−3x u(x) = -3x u′(x)=−3 u'(x) = -3

STEP 5

Differentiate v(x)=6x3−3 v(x) = \sqrt{6x^3 - 3} using the chain rule.
First, rewrite v(x) v(x) as (6x3−3)1/2 (6x^3 - 3)^{1/2} .
v(x)=(6x3−3)1/2 v(x) = (6x^3 - 3)^{1/2}

STEP 6

Apply the chain rule to differentiate v(x) v(x) .
Let g(x)=6x3−3 g(x) = 6x^3 - 3 and w(x)=(g(x))1/2 w(x) = (g(x))^{1/2} .
v(x)=w(g(x))=(6x3−3)1/2 v(x) = w(g(x)) = (6x^3 - 3)^{1/2} w′(u)=12u−1/2 w'(u) = \frac{1}{2} u^{-1/2} g′(x)=18x2 g'(x) = 18x^2
So,
v′(x)=w′(g(x))⋅g′(x)=12(6x3−3)−1/2⋅18x2 v'(x) = w'(g(x)) \cdot g'(x) = \frac{1}{2} (6x^3 - 3)^{-1/2} \cdot 18x^2 v′(x)=18x226x3−3 v'(x) = \frac{18x^2}{2 \sqrt{6x^3 - 3}} v′(x)=9x26x3−3 v'(x) = \frac{9x^2}{\sqrt{6x^3 - 3}}

STEP 7

Apply the product rule to combine the results of u(x) u(x) and v(x) v(x) .
The product rule states:
h′(x)=u′(x)v(x)+u(x)v′(x) h'(x) = u'(x)v(x) + u(x)v'(x)
Substitute u(x)=−3x u(x) = -3x , u′(x)=−3 u'(x) = -3 , v(x)=6x3−3 v(x) = \sqrt{6x^3 - 3} , and v′(x)=9x26x3−3 v'(x) = \frac{9x^2}{\sqrt{6x^3 - 3}} .
h′(x)=(−3)6x3−3+(−3x)(9x26x3−3) h'(x) = (-3)\sqrt{6x^3 - 3} + (-3x) \left( \frac{9x^2}{\sqrt{6x^3 - 3}} \right)

STEP 8

Simplify the expression h′(x) h'(x) .
h′(x)=−36x3−3−27x36x3−3 h'(x) = -3 \sqrt{6x^3 - 3} - \frac{27x^3}{\sqrt{6x^3 - 3}}
Combine the terms under a common denominator:
h′(x)=−3(6x3−3)−27x36x3−3 h'(x) = \frac{-3 (6x^3 - 3) - 27x^3}{\sqrt{6x^3 - 3}} h′(x)=−18x3+9−27x36x3−3 h'(x) = \frac{-18x^3 + 9 - 27x^3}{\sqrt{6x^3 - 3}} h′(x)=−45x3+96x3−3 h'(x) = \frac{-45x^3 + 9}{\sqrt{6x^3 - 3}} h′(x)=9(1−5x3)6x3−3 h'(x) = \frac{9(1 - 5x^3)}{\sqrt{6x^3 - 3}}

STEP 9

The simplified final expression for h′(x) h'(x) is:
h′(x)=9(1−5x3)6x3−3 h'(x) = \frac{9(1 - 5x^3)}{\sqrt{6x^3 - 3}}
Solution: h′(x)=9(1−5x3)6x3−3 h'(x) = \frac{9(1 - 5x^3)}{\sqrt{6x^3 - 3}}

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