Differentiation Questions and Answers

Differentiate y sec 0 tan 0 y sec 0 sec 0 tan e

Differentiate the following function 2x 5 x 4x y

6 Use Logarithmic Differentiation to differentiate to find dy dx given 3 x x 1 x 2 x 1 2x 3 y

Let f v v 5v Determine the critical values of f If multiple such values exist enter the solutions using a comma separated list Ov No critical values exist Submit Question

the following is the graph of f x the function f x has a local A 2 and 1 3 1 and 1 Q0 and 1 Dj 2 only E 1 only hj

Which of the following graphs is the slope field for Plea 111 2111 2 3 4 5 dx cos x O O 11 1 A 543 11 MAX 2 3 4 5 24 04 1 123

f x x Add vertical stretch Then the veritcal Shift up by 3 by 2

Solve the equation by completing the square x 2x 23 0 X X 1 2 6 1 2 6 smaller value larger value

What is the 2023rd derivative for the function f x cosx Show work and ensure your problem solving strategy is clear

Find the local maxima local minima and saddle points if any for the function z Use symbolic notation and fractions where needed Give your answer as point coordinates in the form Enter DNE if the points do not exist local min 3x 27xy 3y local max

Use the product rule to find the derivative of the given function y 3x 2 3x 2

Find the value s of t so that the tangent line to the given curve contains the given poi r t 5ti 2t j 2t k 0 8 8 t 0 Use a comma to separate answers as needed

Find the acceleration function a t if s t 7 7t 4t 3 O A a t 42t 14 OB OC O D a t 21t 4t 14 a t 42t 4 a t 21t 14t 4

For the function f x FULGA 10 x find f x Then find f 0 and f 7

Find all critical numbers of the function f x x 72x Then use the second derivative test on each critical number to determine whether it leads to a local maximum or minimum Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The critical number s is are at x There is no local maximum and no local minimum Type an integer or a simplified fraction Use a comma to separate answers as needed OB The critical number s is are at x There is a local maximum at x There is no local minimum Type an integer or a simplified fraction Use a comma to separate answers as needed OC The critical number s is are at x There is no local maximum There is a local minimum at x Type an integer or a simplified fraction Use a comma to separate answers as needed OD The critical number s is are at x There is a local maximum at x There is a local minimum at x Type an integer or a simplified fraction Use a comma to separate answers as needed OE There are no critical numbers

For the function f x 3ex find f x f 0 f 7 and f 6 f x Type an exact answer in simplified form

4 25 pt Identify the symmetries of the curves r 12 cos 0 and r 4 4 cos 0 Make a table for r values corresponding to 0 values Draw the curves on the same plane Then find the area of the region shared by these curves

3 25 pt Test the function below for local maxima loc al minima and saddle points Find the value of the function at these points f x y x y 2xy y 3y

For the function shown in the graph list the intervals on which the function is increasing the intervals on which it is decreasing and the location of all local extrema AY E List the open interval s on which the function is increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The function is increasing on the interval s Type your answer in interval notation Use a comma to separate answers as needed OB The function is never increasing

8 Find dy by implicit differentiation dx sin xy x y

Explain why Rolle s Theorem does not apply to the function even though there exist a and b such that f a f b Select all that apply X f x cot 7 f a does not equal f b for all possible values of a and b in the interval 7n None of these There are points on the interval a b where f is not differentiable Of a does not equal f b for any values in the interval 7n There are points on the interval a b where f is not continuous X

Find the angle between the planes 6x 3y 10 and 9x 6y 10z 20 The radian measure of the acute angle is 0 Round to the nearest thousandth

Prove the identity sin x sin x x Note that each Statement must be based on a Rule chosen from the Rule menu To see a detailed description of a Rule select the More Information Button to the right of the Rule Statement sin x sin x 1 cotr Rule Select Rule 0 0 cos cot JT 010 3 Osin Osec tan csc

The formula for the power output P of a battery is P VI RI2 where V is the electromotive force in volts R is the resistance in ohms and I is the current in amperes Find the current that corresponds to a maximum value of P in a battery for which V 13 volts and R 0 5 ohm amperes Assume that a 15 ampere fuse bounds the output in the interval 0 I 15 Could the power output be increased by replacing the 15 ampere fuse with a 24 ampere fuse Yes No

Solve the following equation for x x 121 list your answers separated by a comma not using a sign and do not include x in your answer

Find the derivative of the function f x sqrt x 10 oot 3 x

Find the area of the region between the graphs of f x x 3 and g x 2x A 3 B 8 C 6 YA f x x 3 g x 2x D 9 E 5

In Problems 18 19 find the second derivative f x of each of the given functions f x rsin z 18 f x 1 1 dt Answer f x sin r 2 cos r 3 sin 2 3 3 1 sin x 2 3

dy We now solve the resulting equation for dx 2x 20y 2y 20x dy dx 2y 20x dy dx dy dx 0 11

We have determined that d dx x 2x dy 2x 20 x dx y dx 2x 20x dy 20y dx dy xy x Y dx Applying these derivatives and simplifying gives the following result d d d x 20 xy y 20 dx dx 2x 20y d dx dy dx d v 2ydy and 20 0 dx dx dy dx 0 0 dy 1 dx 0

Find all critical numbers for the following function Then use the second derivative test on each critical number to determine whether it leads to a local maximum or minimum f x 2x 3x 36x 6 What is the second derivative of f x t x Find all critical numbers and determine whether they lead to a local maximum or minimum Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The critical number s is are x There is no local maximum and no local minimum Use a comma to separate answers as needed OB There are no critical numbers OC The critical number s is are x There is a local maximum at x Use a comma to separate answers as needed OD The critical number s is are x Use a comma to separate answers There is a local minimum at x There is no local maximum There is a local minimum at x as needed O E The critical number s is are x Use a comma to separate answers as needed There is a local maximum at x There is no local minimum

We have determined that g x F x h g x F x h g x g x 8 X and h x cos x where g x 8 In x and h x sin x Applying the Chain Rule gives the following result

So far we have the following d x 20 xy y 20 d dx dx dx d The derivatives x and dx dx d x 20 dx However we note that the derivatives xy and d dx involve y Recall that the derivative of any term involving y will require the chain rule and include the factor dy 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 We also note t dx d dx xy x dy dx d dx v d 20 can be found using basic derivative rules learned in previous sections Doing so gives the following result dx d dx dy dx

DETAILS Find y and y y y SCALCET9 3 6 029 y x In x

Differentiate the function p t In 6 p t

Find the derivative of the function J 0 tan ne J 0

Find the derivative of the function dy dy y 4e5x 2

Find the derivative of the function f x In 3x 4 4x 5 f x

Find the derivative of the given function y x e 5x 0

Find the derivatives of the following functions g x 2x 7x 4 f x 2x 7x 4 5 g x f x

The average blood alcohol concentration BAC of eight male subjects was measured after consumption of 15 mL of ethanol corresponding to one alcoholic drink The resulting data were used to model th concentration function C t 0 00225te 0 0467t where t is measured in minutes after consumption and C is measured in g dL Round your answers to six decimal places a How rapidly was the BAC increasing in g dL min after 4 minutes g dL min Interpret your answer in the context of this problem The model predicts that the BAC will be Select by this approximate amount after b How rapidly was it decreasing in g dL min half an hour later g dL min Interpret your answer in the context of this problem minutes

Step 2 du We let u 7x and have determined that 7 Applying these gives the following result dx d dx sin u d sin 7x dx 1 du 1 u dx 7 1 1 49 1 7 x 7 Therefore differentiating f x sin 7x gives us the following f x

Differentiate the function f x sin 8 In x Step 1 Recall the Chain Rule in terms of the functions h x and g x which states that if g is differentiable at x and h is differentiable at g x then the composite function F h og define F x h g x is differentiable at x and F is given by the following F x h g x g x We are given the function f x sin 8 In x To write this is the form F x h g x we can let g x 8 In x and h x sin x Step 2 Before applying the Chain Rule we will first find the derivatives g x and h x g x 8 In x g x h x sin x h x sin z

Consider the following equation 4x y 5 a Find y by implicit differentiation y 4x 4x 5 b Solve the equation explicitly for y and differentiate to get y in terms of y t

Find the derivative of the function f x sin 7x Step 1 We recall that the derivative of the function y sin x is given by the following 1 1 x dx We note that for the given function f x sin 7x we will need to combine the formula above with the chain rule as follows where u 7x sin u sin dx U 7x 11 sin x du du 1 0 So our first step is to determine du 1 u dx du dx

Recall the Chain Rule in terms of the functions h x and g x which states that if g is differentiable at x and his differentiable at g x then the composite function F h og defined by F x h g x is differentiable at x and F is given by the following F x h g x g x We are given the function f x sin 8 In x To write this is the form F x h g x we can let g x 8 In x and h x X

Find the derivative of the function y y 5 tan x 1 x

dx x 20xy y2 20 Step 1 Recall that the method of implicit differentiation consists of differentiating both sides of an equation with respect to x and then solving the resulting equation for dy dx We begin by differentiating both sides of the given equation x 20xy y2 20 The beginning of this process proceeds as follows using the sum and difference rules and the constant rule for differentiation d dx d x 20 dx dx d dx x 20xy y Submit Skip you cannot come back dx 1 20 d dx dx 20 d y x 20 dx dx

54 If h r is a differentiable function with h 1 1 h 2 f 2 given that f x eh x 1 h sin r 2 and h T 2 1 evaluate 2

Tutorial Exercise Find the derivative of the function f t 6t sin nt