Differentiation Questions and Answers

Find by implicit differentiation dy dx x x6y6 y 8 dy dx

So far we have the following x 16 xv v 16 dx dx dx The derivatives x and 16 can be found using basic derivative rules learned in previous sections Doing so gives the following result d dx d 16 dx dx d xy and y involve y Recall that the derivative of any term involving y will require the chain rule and include the factor dy We also note the However we note that the derivatives the first of these two derivatives involving y is the product of two differentiable functions so the product rule will be used Finding each of these derivatives gives the following result dx dx dx xy dx dy xy dx dy dx

Let f x g x where gu Step 1 We are given the function f x eg x The first step is to find the derivative f x Let h x ex and note that the func g x Recall the product rule in terms of two differentiable functions h x and g x r x n x g x g x n x To apply this rule we first find the derivative of h x ex h x ex h x d h x g x n x 9 x 9 x n x dx 0 I We now apply the product rule F x h x g x g x g x h x 0 90x g x

Find by implicit differentiation dy dx dy dx ex cos y x y

Find the derivative of the function f x f x 3 1 2x 3

a The curve y x 1 x2 is called a serpentine Find an equation of the tangent line to this curve at the point 2 0 40 y b Illustrate part a by graphing the curve and tangent line on the same screen y 2 2 655 0 5 1 0 WebAssign Plot 1 0 0 5 0 5 y 2 2 4 4 2 O y 1 0 0 5 6 5 2 4 4 2 y 1 0 0 5 0 5 Sten

Find the derivative of the function F t e8tsin 2t F t

dy Find for the equation 4x 5y5 4 using implicit differentiation dy 11

DETAILS Find the derivative of the function y tan 5x y

Let fand g be the functions in the table below g x 2 3 1 X 1 2 3 f x 3 1 2 a If F x f f x find F 2 F 2 b If G x g g x find G 3 G 3 f x 4 5 7 g x 6 7 9

dy Find by implicit differentiation dx dy dx 8 x 9 y 24

The velocity of a particle in ft s is given by v t 6t 5 where t is the time in seconds for which it has traveled Find the time at which the velocity is at a r OA 6 s OB 5 s C 3 s O D 2 5 S

Verify that wxy Wyx for w In 5x 4y

Find f f f 2 2 f x y z 3x 3y z www fx Type an exact answer using radicals as needed

4 Listen Determine the derivative of the following function Do not simplify 3A f x 3x 3 4x 2 3x 1 33

The derivative of y 34x is Oyl 342 In 4 Oyl 4 342 In 3 Oy 4 342 In 4 4 342

12x1 3 10x5 7 dx 12x1 3 10x6 7 10x6 7 dx Type Type an exact answer

Find the real and complex zeros of the function f x x 2x 6x 12x 9x 18 Express your answer as a list separated by commas Do not use a plus minus sign t in your answer instead list the zeros separately Do not include x in your answer Provide your answer below

The polynomial f x given below has I as a zero Find all the other zeros of f x Provide your answer below f x x 5x 9x 5

P x 37 60 x5 259 433 x4 24 6 x3 5489 24 x 20083 60 x 158

Divide 10x4 10x 15x260x 47 by 5x 5

Using the function f x 3x 7x3 3x 5x 4 and Descartes Rule of Signs what are possible combinations of positive real negative real and non real complex zeros Select all that apply 3 positive real 1 negative real 0 non real complex 4 positive real 0 negative real 0 non real complexi 0 positive real 0 negative real 4 non real complex 2 positive real 1 negative real 1 non real complex 1 positive real 1 negative real 2 non real complex

te an equation of the form y a sin bx or y a cosbx to describe the graph below MA B 00 S

g x g x x 3 x ex Enhanced Feedback X Please try again using the Product Rule which states d Dafy f x a x f x d a x ald d CEL

DETAILS LARCALC11 3 3 071 MY NOTES ASK YO A differentiable function has one critical number at x 3 Identify the relative extrema of f at the critical number f 2 3 5 and f 4 2 O 3 f 3 is a relative maximum 3 f 3 is a relative minimu

Differentiate y y 3x 2 4x 5

Find the limit lim X40 sin 4x 3x

Differentiate f x f x x6 ex x6 ex

Differentiate the function s t s t 1 t To H t

Differentiate y 2x sin x Solution Using the Product Rule and the formula sin x cos x we have dy dx 2x3 d dx 2x d dx sin x 1 d dx sin x

a tan x sin x cos x sec x 1 sin x O sec x tan x O cos x sin x sec x tan x 1 cos x O sec x b sec x O cos x sin x O sec x sin x cos x 1 cos x O sec x tan x cos x sin x 1 sin x O sec x tan x O cos x sin x tan x 1 sec x sin x cos x cot x 1 CSC x O cos x sin x tan x 1 sec x cos x sin x cot x 1

Differentiate g x x x 3 x ex

The campus bookstore has estimated that its profit in dollars from selling x hundred basketball conference championship t shirts is given by the equation shown below P 43x 585x 524 The demand is currently 800 t shirts but euphoria over the championship is subsiding the demand s dropping by 100 t shirts per day How is the profit changing with respect to time n GECES

The total stopping distance T of a vehicle is shown below where T is in feet and x is the speed in miles per hour T 2 5x 0 5x2 Approximate the change and percent change in total stopping distance as speed changes from x 20 to x 21 miles per hour Round your answers to one decimal place dT ft dT

Use the information to find and compare Ay and dy Round your answers to four decimal places y x 2 x 1 Ax dx 0 01 Ay dy

Find the differential dy of the given function y csc 8x dy dx

Find the tangent line approximation 7 to the graph of f at the given point Then complete the table Round your answers to four decimal plac f x csc x 3 csc 3 T x X f x T x 2 9 2 99 3 3 01 3 1

ind fog x and gof x and the domain of each f x x 11 g x 3x 5 fog x Simplify your answer

Two airplanes are flying in the air at the same height Airplane A is flying east at 200 mi h and airplane B is flying north at 250 mi h If they are both heading to the same airport located 120 miles east of airplane A and 50 miles north of airplane B at what rate is the distance between the airplanes changing Enter an exact answer Provide your answer below The distance between the two planes is decreasing at a rate of miles per

Which function has an actual rational zero of 1 Select the correct answer below O g x x 3x x 7 g x x 7x x 7 O g x x 3x x 8 O g x x 7x x 8

The body mass index BMI is a number that can be calculated for any individual by using the equation B 703W where weight w is in pounds and height h is in inches Use this information to h complete the following a Calculate the BMI for a person who weighs 214 pounds and is 74 tall BMI Round to the nearest whole number as needed b Calculate B ch 20 c Calculate 28 B dh 38 w

Therefore we need to differentiate each side of our area equation with respect to t using the product rule Doing so gives the following result where Step 4 dw dt dA dt dA dt dA dt Step 3 We are given that the length is increasing at a rate of 4 cm s and the width is increasing at a rate of 9 cm s Therefore we have the following dl dt d lw dt 9 2 6 4 Furthermore we are given that the length is 8 cm when the width is 6 cm In other words when 8 we have the following W 6 dl Substituting the values 4 dt 1 1 w dt 9 dw dt W dl dt dl W dt dw dt 9 1 8 and w 6 into the derivative and simplifying the result gives the following Therefore the area of the rectangle is increasing at the following rate in cm s dA dt is measured in cm2 s

Step 2 We need to determine how fast the area of the rectangle is increasing In other words we are looking for a rate of change of the area In this problem the volume the length and the width are all functions of the time t where t is measured in seconds The rate of increase of the area with respect to time is the derivative dA The rate of increase of the length and width with respect to time are the derivatives dl dt dt respectively Therefore we need to differentiate each side of our area equation with respect to t using the product rule Doing so gives the following result where dA d lw dt dt dA dt dw dt C dl dt dA dt dw and 1 dt is measured in cm s

Determine if the graph is a function Using the vertical line test is this the graph of a function No Yes 101 81 8

A street light is mounted at the top of a 15 ft tall pole A man 6 feet tall walks away from the pole with a speed of 4 ft s along a straight path How fast in ft s is the tip of his shadow moving when he is 30 feet from the pole ft s

Tutorial Exercise Find the linearization L x of the function at a f x x x 2 a 2 Step 1 Recall that the linearization of f x for x a is given by L x f a f a x a For f x x x 2 we have the following f x I

Consider the following inequality problem 6w 7w 6 or 1 5w 51 Step 3 of 4 Using your answers from the previous steps solve the overall inequality problem and express your answer in interval notation U decimal form for numerical values Answer 2 Points Key Keyboard Sho

Tutorial Exercise Duration A population of a particular yeast cell develops with a constant relative growth rate of 0 4775 per hour The initial population consists of 3 4 million cells Find the population size in millions of cells after 6 hours Step 1 dP Since the relative growth rate is 0 4775 per hour then the differential equation that models this growth with P in millions of cell and t in hours is dt

V Consider the following inequality 2z 4 5 Step 1 of 2 Solve the linear inequality for the given variable Simplify and express your answer in algebraic notation Answer 2 Points If all real numbers satisfy the inequality select All Real Numbers If no real number satisfies the inequality select No Solution Ke Keyboard She Selecting one of the alternative options will replace the entered value s with the selected value Otherwise the entered answer is used O No Solution

Use the graph of f to complete the following table X x 0 X 0 x 0 x 0 f x f x f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 Sketch the graph of f and f on the same coordinate axes X i