Let f (x) sin? æ. Define F (x) So f (t) dt . What is F" (E) ?

Q: O Find the derivative of the function. F(x) = | cos (*) dt

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Q: Is there a function f(x, y) such that fx : one. If not, explain the reason = e* cos y and fy : = e*…

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Q: d Find the value of f(t) if f(t) sin (Vx) dx. dt f(t) =

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Q: a. Evaluate the following derivative d tổ tan(t) dt. sin(x) dx

A: We will use the Newton Leibniz formula to evaluate the differentiation

Q: Find F'(x) given that F(x) = sin (t2) dt

A: Derivative of integration

Q: Find the particular antiderivative of the following derivative that satisfies the given condition.…

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Q: Apply the Fundamental Theorem of Calculus to get the following derivative. d. In( t?+1) dt dx 1

A: Given, fx=ddx∫1x3lnt2+1dt To Find: f(x) by applying fundamental theorem of calculus.

Q: Apply the Fundamental Theorem of Calculus to get the following derivative. d In (t² + 1) dt dx 1

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Q: d Find the value of f(t) if f(t) : sin(Vx) dx. dt. f(t) %3D

A: here use the fundamental theorem of calculus

Q: Evaluate f(x) dx, where f(x) = 13 sin(x), 0< x < T. f(x) dx =

A: The solution is given as

Q: Find d2x/dt? as a function of x if d 6x sin x. dt? ||

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Q: tan x dt+ 1+ t? (1+ f')* 1 -dt=D1 1/e Ve prove LHSis equal t RHS

A: This question is related to integration. We solve this question we divide this question in two part…

Q: g(x) = sin(t? + 1)dt

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Q: dy Find the derivatives dx of the following functions csin x Vt2 + 1 dt 2

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Q: Let F(x) = sin 3t dt, please find F'(0)

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Q: Use the Product Rule to compute the derivative: d. Find dt (e + 1)(t + 9)) at t = 13

A: Topic :- Derivative

Q: Find dy/dt if y = tan^−1 (1 /√t)

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Q: d Find the value of f(x) if f(x) : cos (t) dt. dx. COS f(x) =

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Q: What is the value of f (5) if f (t) sin (Ta) dt :? 1

A: Given, ∫1xfttdt = sinπxx   We have to find the value of f5

Q: Evaluate dx f(t) dt and * f(t) dt, where a and b are constants.

A: To evaluate: ddx∫axft dt and ddx∫abft dt, where a and b are constants. Solution: The fundamental…

Q: 2x g(x) = |VI+t? dt 1+t2 dt %D

A: To find the derivative

Q: Find G(1), G'(0), and G'(π/4), where G(x) =

A: Here we have;  G(x)=∫1xtant dt

Q: Evaluate the following derivatives.

A: Given, ∫x2exsin3t dt We have to find ∫x2exsin3t dt ∫x2exsin3t dt=∫x2exsin2t sint dt ∫x2exsin3t…

Q: Find the derivative of y with respect to t. y=e(cos t+ln t) dy/dt=???

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Q: Suppose f(x) = | sin(t?) dt. Find f'(x). x2

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Q: Find the derivative (dy/dx) of the following functions: h) у%3D Зx7 —7х3 + 21x2 - i) y = (x² +…

A: Since you have asked multiple questions in a single request, we will be answering only the first…

Q: Given: y=fso(x-1) dt - dy =D{(x). dx show that

A: We will prove the following result using Leibnitz rule.

Q: If F (x) = (t2) dt , then F' (4) = cos 0,

A: DUIS rule is used

Q: Use the Quotient Rule to compute the following: d ( t2 +1 at t = -4 %3D dt \ t2 – 1 The derivative…

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Q: If F(x)= J. x) = [", "+!dt then F'(x) dt , then F'(x)= /3 sint

A: If Fx=∫π3xt2+1sintdt, then evaluate the value of F'x.

Q: Given that f has a continuous derivative, compare d d f(t) dt [f (t)] dt. dt to dx

A: Given that, f has a continuous derivative, compare  ddx∫axf(t)dt to ∫axddtf(t)dt

Q: Compute the derivative: d I/ In(t° + 1) dt dx

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Q: Use the method of first principles to show that d/dt(lnt)=1/t

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Q: Use the Quotient Rule to compute the following: d t2 +1 dt t2 – 1 The derivative is:

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Q: Find F'(x). 2x F(x) = cos(t*) dt

A: If f(x) is continuous over an interval [a, b], and the function  F(x) = ∫au(x)f(t) dt then F'(x) =…

Q: dy dy For the function dt 7 cos t, the third order derivative dt3 is %3D

A: derivative with respect to t first oder derivatinve dydt is ddt(y). (which is given in the…

Q: Evaluate dt dW at t=0 for the function W = e' – In x , x = t², y= In t

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Q: a) If d'r = 3, when t = 0 we have dt? dx 0 and x=0, what is x as a function of dt t?

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Q: Use the Quotient Rule to compute the following: d/dt(t^2+1/t^2-1) at t=4   The derivative is:

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Q: Find the second of the finchion, ç x) = cos e derivative f (x) (x %3D %3D

A: Consider the given function: f(x)=xcosx The first derivative is calculated using the product rule.…

Q: cos(x) If f(x) = ["(@) "g(x) V1 + t3 dt, where g(x) = S [1 + sin(t²)]dt, find f'(;). 1 %3D

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Q: Use Part 2 of the Fundamental Theorem of Calculus to find the derivatives. d/dx ∫_0^x▒〖e^√t dt〗…

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Q: f(t) dt t sin(rx), -? What is the value of f(5) if

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Q: Consider the function g(x) = 5 S cos t dt – x?. Then g' (x) is

A: Given gx=12∫x4x2costdt  -x2

Q: Find fr and fy- f (x, y) = 8e dt

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Q: g(x) f(x) - sin(2²) dt, find f(x). = J

A: To Find: f'(x)

Q: Suppose y = (sq. root of) 2x + 1, where x and y are functions of t. (a) If dx/dt = 9, find dy/dt…

A: y=2x+1 Differentiating with respect to t, we get dydt=122x+1·d2x+1dt…

Q: Si f(t) dt = x² – 2x + 1. Find f(x). 79. Suppose that

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Q: Find the particular anti derivative that Satisfies the following conditions dy 4x +3 ; y(1) = 8 dx…

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Q: f(x) = f(x) +² dt where g(x) = [cosx[1 + sin t²]dt, find f'(-)

A: It is known that: ∫g(t)h(t)f(x)dx=h'(t)f(h(t))-g'(t)f(g(t)). We need to calculate f'π2 using the…

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.