Use mathematical induction to prove that the derivative of f (x) = xn equals nxn−1 whenever n is a positive integer. (For the inductive step, use the product rule for derivatives.)

Q: Find the derivative of the following function F(x) = (21 – 1)° dt %3D using the Fundamental Theorem…

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Q: Find the derivative of the composite natural exponential functions 1. f(x) = In( 2. g(x) = In(v³ +1)…

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Q: compute the derivative. c(t) = 1-li – e2'k - e2

A: Given  ct=1ti^-e2tk^

Q: Proof the n + 1 derivative of xk is zero, where k <n

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Q: Use the u-substitution to find the anti-derivative of m'(x)= 1/xln(x)

A: solution :- antiderivative is the integration  So put ln(x)  = u Then (1/x )dx  = du And then…

Q: Use part one of the fundamental theorem of calculus to find the derivative of the function. g(x) =…

A: Solution: The objective is to find the derivative of the given function

Q: 2 Using the definition of derivative, prove the following: (a) f'(x) = x if f(x) = x %3D

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Q: F(x)= ]7+1 3x t dt 2 F'(x) = %3D

A: We will find the derivative using Fundamental Theorem of Calculus.

Q: 1. Compute the first, second, and third derivative of a. f(x) = x² In(x+ 1)

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Q: 18. If f (x) = es*, find (f-1)'(1) using Theorem 7 (Derivative of the Inverse Theorem).

A: We have to find derivative

Q: Use the Product Rule twice to find a formula for (fg)" in terms of f and g and their first and…

A: Product rule in derivative:  [A(x).B(x)]' = A(x).B'(x) + A'(x).B(x)

Q: 88. Use the definition of the derivative to prove the following special case of the product rule: d…

A: We can use the product rule of differentiation when we have the product form of two functions that…

Q: use the Power Rule to compute the derivative.

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Q: 2. g'(x) where g(x)= x² +7x-5 %3D

A: ddxxn = n·xn - 1, and ddxC =0

Q: Can you help with this problem step by step?

A: Given,

Q: write the function as a composite f (g(x)) and compute the derivative using the Chain Rule. y =…

A: Given : The function is y = cos ( x3 ) .

Q: Use derivatives to find two positive numbers a and b such that a^2 + b^2 = 1 and such that 2*a+3*b…

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Q: Use the General Power Rule, Exponential Rule, or the Chain Rule to compute the derivative.

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Q: 68. Use the definition of the derivative and the definition of the number e to prove that f (x) = e*…

A: The given function is fx=ex. Obtain the derivative as follows.

Q: 2. g'(x) where g(x)=x² +7x-5 %3D

A: In this question we need to find a derivative of the given function.

Q: Let f (x) = e10x. %3D Find a formula for the n-th derivative of f, for n 2 1.

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Q: Compute the derivative function f' (x) algebraically. f (x) = –x² – 2x O f' (x) = 2x – 2 O f' (x) =…

A: Derivative of xn= nxn-1

Q: use the Power Rule to compute the derivative.

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Q: Find the derivative of the following function F(x) = | (21 – 1)° dt using the Fundamental Theorem of…

A: ∫abf(x)dx=-∫baf(x)dxAccording to fundamental law of calculusddx∫0f(x)F(t)dt=f'(x)*F(f(x))

Q: Starting from the definition of derivative prove that if f(x) = 1 then f'(a) = a2

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Q: if f(x) = x²√√In x² P x A. f'(x) = . find the derivative of this function f'(x) 1-In x² x²√√In x²…

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Q: if f(x) = x²√√In x² A. f'(x) = B. f'(x) = c. f'(x) = D. f'(x) = find the derivative of this function…

A: Given,            fx=x2lnx2xTo find : Find the derivative of the given function.

Q: Use the Fundamental Theorem of Calculus to find the derivative of F(x). Simplify your answer. COs 1…

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Q: Use the Fundamental Theorem of Calculus to find the derivative. Vx-x1-t2 dt d dx

A: The Fundamental Theorem of Calculus says: ddx∫ag(x)f(t) dt=f(g(x))·g'(x)...(1)

Q: Find the derivative of the following function x4 F(x) = (2t – 1)3 dt using the Fundamental Theorem…

A: Consider the given function in the form of the definite integral for f(t).

Q: Find the derivative of the function ffor n = 1, 2, 3, and 4. sin x fx) = - (X), I n =4 Use the…

A: Solve for the derivative

Q: 4. Find all derivatives of f(x) = x*, where k is some unknown natural number.

A: f(x) = xk and we need to find all the derivatives of f(x)

Q: Obtain the first and second derivative of the following real-valued functions 1. ƒ : R → R, such…

A: Solution :1.fx=23x3-2xfirst derivative ,f'xf'x=29x2-2Second derivative…

Q: Compute the derivative of f(1) = In(x In(x² In(r³))).

A: The given function f(x)=lnx lnx2 lnx3

Q: Find the derivative of the function. 92-4x + h(x)= e 18x – 4x2 –1 92-4z++ x2 O b. 9x - 8x² – 1…

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Q: Find the derivative of the following function (2t – 1)* dt F(x) = using the Fundamental Theorem of…

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Q: d) Hence form a conjecture for the derivative of functions in the form y = (ax + b)². e) Prove your…

A: As you have mentioned I am answering Question (d) and Question (e)

Q: Find the derivatives of the function f for n = 1, 2, 3, and 4. Use the results to write a general…

A: Given:fx=cos xxn for n=1,2,3,4Formula:ddxuv=vu'-uv'v2

Q: Compute the derivative function f' (x) algebraically. f (x) = -x? – 2x O f' (x) = -2 – 2 %3D O f'…

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Q: 2. Using the definition of derivative, prove the following: 1 (a) f^(x)= if f(x) = Vx %3D

A: fx=x3

Q: Use the ħndanental Theorem Ot calculus part I to find the derivative Of hx)=

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Q: Compute the derivative function f' (x) algebraically. f (x) = -5x3 +7 O f' (x) = -15x O f' (x) =…

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Q: = [₁²₁ Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of g (r) : = x² + 4dx.

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Q: Use the u-substitution to find the anti-derivative of h'(x)= xcos(2-3x2)

A: h'(x)=xcos(2-3x2) Formula ∫cos (x) dx= sin (x)+C

Q: The general anti-derivative of the function f (x) - In(x) is given by x In( x)+ C Inx)-x+ C x In(x)…

A: The integral (anti derivative) of fx=lnx can be evaluated by using integration by parts. The…

Q: Show that the function f(z) does not admit an anti-derivative in the domain N = C\{0}. Does it admit…

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Q: Find an expression for the nth derivative of f(x) = xe² by computing the first few derivatives,…

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Q: Let f (x) = x sin x and g(x) = x cos x. (a) Show that f ,(x) = g(x) + sin x and g ,(x) = −f (x) +…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: