Which of the following is the correct limit defihition for the derivative of g(x) = + 4? Select the correct answer below: (x+h) +4 - ( +4) lim (x+h +4- ( +4 lim (x+h) +4- (+4 lim Ha -(+4) +4

Differentiate the following function. y=x(x³ +5) ³ Choose the correct setup below to differentiate the function. [x(x5 +5)³] = (x)•. J :(x5 + 5)³) + ((x³ +5)³).) dx d « (x5 + 5)³] = « - -(x* + 5) ³) - ((x³ +5)³).) dx dx [x (x* +5)³]=2• [x(x° +5) ³].[x (x* + 5) ³] dx dx d d Ix(x5 +5)³]= x) • 3 (a5 +5) + ((x³ +5) ³) · x) d [x (x5 + 5)³] = (x) • ) + ( (x5 + 5) ³) • -c() + ( (x5 +5) ³) • (x5 + 5) ³) dx dx dx d [x(x* + 5)³ : dx

Evaluate Function  g(3,4) when g(x,y)=x-6y/x^2+y^2 a. -25/21 b. -5/21 c. -21/25 d. -21/5

Consider the function f(x)=x/x^2+12x+32. Determine the intervals on which f is increasing and decreasing. Your answer should either be a single interval, such as "(0,1)", a comma separated list of intervals, such as "(-inf, 2), (3,4)" , or the word "none".

If we need to calculate the derivative of a function at a point, there are two ways we can think about doing this. For example suppose f() = x – x, and we need to determine the value of f(4). One option is to just calculate the derivative at that point by plugging the point into the limit definition, like this: f(4 + h) – f(4) ( (4 + h)² – (4 + h)) - (4² – 4) f'(4) = lim (16 + 8h + h2 -4 – h) - (16 - 4) lim 7h + h2 = lim h(7+ h) lim h = lim h→0 h lim 7+h = 7. || h h h Another option is to calculate f'(x) in terms of x, and then plug in a value for x at the end, like this: f(x + h) – f(x) (2+ h) – (x + h)) - (2² – a) (22 + 2xh + h² – z – h) – (22 – 2) lim f'(x) = lim = lim 2xh + h2 - h h(2x + h - 1) = lim h 0 h h = lim = lim h h h that is to say, f'(x) = 2x – 1, and therefore f'(4) = 2(4) – 1 = 7. Notice that we get the answer f'(4) = 7 both ways. The second approach might look somewhat more complicated at first, but it turns out to be much more efficient if we ever need to know the…

When applying derivative rules, we sometimes need to rewrite our function. In this problem, you are not computing derivatives. You are only reviewing the skills necessary to rewrite the expression in an equivalent form. (a) Which is equivalent to 1 10 10x-3 -3 X (c) Which is equivalent to 10x + 3x 1 1 10x3 (b) Which is equivalent to ¹x10 ? O x11/10 O x10/11 O 10x + 3 ? 10x² + 3? X (d) Which is equivalent to √x(x-10√x)? O x3/2 - 10x O x3/2 - 10x1/2

Find the derivative of f(x) =8x – 5x+2 using first principles (i.e., use the definition of the derivative as a limit)