What is the derivative of the function f(x)= (x^2+3)/(sin(x))?
f(x) is the quotient of two simpler functions so we can use the quotient rule to find the derivative. The quotient rule says that if f(x)= g(x)/h(x), then f'(x)=(g'(x)h(x)-h'(x)g(x))/(h(x))^2. In this example, g(x) = x^2+3 and h(x) = sin(x) Using basic derivatives, g'(x) = 2x and h'(x) = cos(x) So using the quotient rule, f'(x) = [(2x)(sin(x))-(cos(x)(x^2+3)] / (sin(x)^2)
What is the slope of any line parallel to the line 13x + 4y = 13 in the coordinate plane?
Parallel lines have the same slope, so we just need to find the slope of the given line, 13x + 4y = 13. The easiest way to find the slope of that line is to get it into slope-intercept form (which just means that we solve the equation for y). If we are trying to solve for y, the first step is to subtract 13x from both sides. After doing that we get 4y = 13 - 13x. Then to finish solving for y, we have to divide both sides by 4. We get y - 13/4 - 13/4x. The slope is just the coefficient with the x variable, so the slope is -13/4. Therefore the slope of any parallel line would be -13/4.
For every dollar Emily spent on groceries, Susan spent 25 cents less. Emily paid 12.50$ more than Susan. How much did both of them spend together at the grocery store?
Let's let variable x stand for how mush Emily spent, and let y stand for how mush Susan spent. The problem tells us that "For every dollar Emily spent on groceries, Susan spent 25 cents less." Another way to think of that is that Susan spends 25% less than Emily, which means that she pays 75% of the price that Emily pays. So y=0.75*x. We also know that "Emily paid 12.50$ more than Susan." So x=y+12.50. Substituting the first equation into the second, we get x=0.75x + 12.50. After we subtract 0.75x from both sides, we get 0.25x= 12.50. Multiply both sides of the equation by 4 to get x=50. If we substitute x=50 back into either of the original equations, we get y= 37.50. So Emily spent $50, Susan spent $37.50, and altogether they spent $87.50.