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mote1985 [20]
3 years ago
7

How would you solve this problem?

Mathematics
1 answer:
klio [65]3 years ago
3 0

Answer:

a) 6x²/2x³-4

b)  2ln (2x^3-4)+ C

Step-by-step explanation:

a) Given the ln(2x³-4). We will use the chain rule in differentiating the function

If y = ln(2x³-4);

u = 2x³-4; du/dx = 3(2)x³⁻¹

du/dx = 6x²

y = ln u; dy/du = 1/u

According to chain rule, dy/dx = dy/dy*du/dx

dy/dx = 1/u * 6x²

dy/dx = 1/2x³-4 * 6x²

dy/dx = 6x²/2x³-4

Hence, the derivative of the given function is 6x²/2x³-4

b) Given an integral function \int\limits {\frac{12x^2}{2x^3-4} } \, dx, the integral problem can be solved using integration by substitution method as shown below;

From the question, let y = 2x³-4... 1, dy/dx = 6x²

dx = dy/6x² ... 2

Substituting equation 1 and 2 into the question given;

\int\limits {\frac{12x^2}{y} } \,\frac{dy}{6x^2} \\\\= \int\limits {\frac{2dy}{y} } \\\\= 2 \int\limits {\frac{dy}{y} }\\\\= 2lny + C\\substituting\ y = 2x^3-4\ into\ the \ resulting\ function\\\\= 2ln (2x^3-4)+ C

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<span>The real problem, in certain fields, is usually to show that for all numbers in that field, there exists an additive inverse. </span>

<span>Therefore, if you tell me that you have a number, and its additive inverse, and you plan to add them together, then I can tell you in advance that the sum MUST be zero. </span>

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<span>--- </span>

<span>A rational number is simply a number that can be expressed as the "ratio" of two integers. For example, the number 4/7 is the ratio of "four to seven". </span>

<span>It can be written as an endless decimal expansion </span>
<span>0.571428571428571428....(forever), but that does not change its nature, because it CAN be written as a ratio, it is "rational". </span>

<span>Integers are rational numbers as well (because you can always write 3/1, the ratio of 3 to 1, to express the integer we call "3") </span>

<span>The additive inverse of a rational number, written as a ratio, is found by simply flipping the sign of the numerator (top) </span>

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<span>and if you ADD those two numbers together, you get zero (as per the definition of "additive inverse") </span>

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<span>If you need to "prove" it, you begin by the existence of additive inverses in the integers. </span>
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<span>Next, show that this (in the integers) can be applied to the rationals in this manner: </span>

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<span>(0)/7 = 0/7 = 0 </span>

<span>Since this is true for ALL integers, then it must also be true for ALL rational numbers.</span>
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