Differentiate: y = (5x +12)^2
This problem is asking us to take the derivative of the equation. This particular problem requires that we use the chain rule because it has an inner and outer portion. This means we will take the derivative of the outer portion first, leaving the inner portion alone, and then multiply that by the derivative of the inner portion. The outer portion of this equation is the squared portion, so let's proceed. 1. We begin by taking the derivative of the outer portion (or the squared portion) of the equation and leaving the 5x + 12 as it is. To take the derivative, we take the number the equation is raised to, multiply it in the front, and subtract 1 from the number the equation is being raised to. Like this: (2)(5x + 12)^(2 - 1) = (2)(5x + 12)^(1) = (2)(5x+12) = 10x + 24 2. Now we will take the derivative of the inner portion of the equation, or 5x + 12. To do this, we will think of x as being raised to the 1st power. Like this: 5(x^1) + 12 = (1)(5)(x^(1 - 1)) + 0 = (1)(5)(x^0) + 0 = (1)(5)(1) + 0 = 5 *Note: the + 12 portion becomes 0 because there is no x by the 12. When there is no x, taking the derivative of a constant, or plain number, causes it to become a 0. 3. Now we will multiply the two portions from steps 1 and 2. Like this: (10x + 24)(5) = 50x + 120 So our final answer is: (dy/dx) = 50x + 120
7(-3x - 5) - (x - 3) = -4(7x + 5) + 16
In this problem we want to find what x would be equal to. In order to do that we have to get all the x's on one side. But some of the x's are being multiplied by other numbers, so first we have to distribute, or multiply. For the left side of the equation: 1. For 7(-3x - 5) we would multiply 7 by -3x and multiply 7 by -5 then we would add those two together. Like this: (7)(-3x) + (7)(-5) = -21x - 35 2. Next we would distribute the minus sign in front of (x-3). We do this by multiplying a -1 by (x-3), or in other words, multiply -1 by x and multiply -1 by 3, and then add those two together. Like this: (-1)(x) + (-1)(-3) = -x +3 3. Now we can add step 1 and step 2 together to get the left side of our equation: -21x -35 -x +3. This simplifies down to: -22x -32. For the right side of the equation: 4. For -4(7x + 5) we would multiply -4 by 7x and multiply -4 by 5 then we would add those two together. Like this: (-4)(7x) + (-4)(5) = -28x -20 5. Then we would add 16 to what we got in step 4. Like this: -28x -20 + 16 which simplifies down to: -28x - 4 Now if we take the two simplified sides of our equation, our equation looks like this: -22x -32 = -28x - 4 6. Now we will combine the x terms. So we can add 28x to both sides to get 28x to cancel out on the right. So now we have -22x -32 + 28x = - 4, which simplifies down to 6x -32 = - 4. 7. Next we will combine the terms without an x. So we will add 32 to both sides to get 32 to cancel out on the left. So now we have 6x = - 4 + 32 which simplifies down to 6x = 28. 8. Now the x needs to be alone, so we will divide both sides by 6. Like this (6x/6) = (28/6). The 6's cancel on the left leaving us: x = (28/6) 9. We can now reduce by dividing by 28 and 6's common factor, which is 2. Which leaves us our final answer of: x = (14/3)
What does SWOT stand for in SWOT Analysis? Explain.
S - Strengths, W - Weaknesses, O - Opportunities, T - Threats. SWOT Analysis is a tool used to determine the internal strengths and weaknesses, and external opportunities and threats facing a company. For example, a small, local grocery store may have a strength of having a "home-town" feel, a weakness of not having all the items a store like Walmart would have, an opportunity to offer online shopping and pick-up/delivery, and a threat of going out of business due to their customer base going to a larger grocery store.