Use the definition of the derivative to show that the derivative of y = f(x) = 2x² is f'(x) = 4x. %3D fu)=2x2

Q: Find the 5th derivative of the function y=x2sin2x at x=0.

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Q: Calculate the derivative with respect to x of the other variable appearing in the equation. exy =…

A: Given, exy=siny2

Q: By using the definition of the derivative, find the for: - dx 1- f(x) = Vx

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Q: 14. If x2 + y2 13, find at (2, 3). dx2 %3D

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Q: Find the derivative using the long method y = vx + 2

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Q: 45. y = cos vsin(tan x)

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Q: (x²+1)* V1-x y = In

A: Use the chan rule of derivative then simplify the result.

Q: Compute the derivative of y = ln ( – V1 – 2 cosh x (x° + 2) |

A: To compute the derivative of,  y=ln1-2 cosh x .x3+2x.x-1.

Q: Find the derivative of the function y = arctan x + x 1+x²

A: Given: y=arctanx+x1+x2Recall the formulas:ddxarctanx=11+x2     ..........1ddxuv=vdudx-udvdxv2…

Q: What is the first derivative of f(x) = ln 3z²+7z ?

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Q: Find the derivative of y with respect to x. y = sec -1 (3x2 + 1), X>0

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Q: Find the derivative of the function. y3 (4x* - 7x +7)* dy dx (Type an expression using x as the…

A: It is given that y=4x4-7x2+74.

Q: Find the derivative of the function. y= log sv7x +3 dy dx I3D

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Q: 2. In y = 2" + x cosh 2x

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Q: Estimate the derivative of f(x) = sin x at x = π/6.

A: Given,       f(x) = sin x at x=π6

Q: Find the derivative of 3 In xy + sin y = x².

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Q: Find the second derivative of y by implicit differentiation from the equation * 4x² + 8y² = 36 64x²…

A: Given that 4x^2+8y^2=36 To determine the second order differential eqaution

Q: Find the derivative of the function. x + 6 X-6 y' = y = In

A: I have appended the formula used in the solution part of the question.

Q: Obtain the derivative of the function y (3 sen(4x))° ( tgʻ(2x)) y = (sec(2x))* – 1

A: This question is based on Trignometric application and diffrentiation of function.

Q: calculate the derivative with respect to x. ln x + ln y = x − y

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Q: Find the derivative of the function

A: Given:y=arctanx+x1+x2Differentiate formula:ddxarc tanx=11+x2ddxuv=vdudx-udvdxv2

Q: Find the derivative of y with respect to x.

A: Assume z=x^5+1. Then found derivative dy/dx using the chain rule.

Q: Compute the derivative of the fo f(x)= 2x+3x ex

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Q: Find the derivative of the function. y = (( 1- cos 2x ) /( 1+ cos 2x ))4

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Q: (3x+4` x² +5, y = tan

A: We know that d(tanx)dx = sec2(x) Now we will apply chain rule , Here we have    y = tan3x+4x2 +…

Q: Find the derivative of the function. y = 3 tan-(x - V1+ x?

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Q: Find the derivative of the function. y = tanh6(9x) %3D y' =

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Q: Find the derivative of the function y = x² cosh²3x

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Q: Calculate the derivative.

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Q: Find the derivative of the function y= arctan x +x/1+x2

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Q: Find the second derivative of the function.

A: Find the second derivative of the function.           f(x)=x sinx

Q: Find the derivative of the function.

A: The given function is y=x2-sin2x4

Q: Find the local extrema of following equation with the second derivative test. y = 3x³ + 2 x³

A: Given: y=3x3+15x5 To Find: The local extrema of the given function using second derivative test.

Q: Find the derivative of the function defined by the equation x+y=arctan(xy).

A: To find the derivative of the given function.

Q: Find the 3" derivative of the function y=f(x) given by the equation y' = 2x.

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Q: Find the derivative of the function y = arctan x+ 1+x?

A: To find: The derivative of the function . Calculation: Find the derivative of the function.

Q: The derivative of y = -ln(cos(ex + 1)) is

A: Our Aim is to find the derivative of y=- ln cos ex+1 -(i)

Q: Find the derivative of: x²y = (y + 2) + ry sin(x)

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Q: Determine the derivative of y. y = eV14x+6 %3D

A: Please refer the attached image for complete solution.

Q: Find the derivative of the function.

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Q: What is the second derivative of y = sec “(3 x) ?

A: For the question 1, none of the options match. If the question had asked for first derivative of y…

Q: Find the derivative of y with respect to x for y=2x   ???

A: y=2x

Q: What is the first derivative of f(æ)= ln(3z²+7z )? 5z

A: Given  f(x)=ln3x2+7x5x We know that derivative of ln(x)=1/x…

Q: find the derivative of y with respect to the givenindependent variable. y = 2x

A: y=2x

Q: find the derivative of the fuction f(x)= 5sinx/3x+cosx

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Q: Find the derivative of y with respect to x.   y= tan-1 *square root (16x2-1) +csc-1*4x ,   x greater…

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Q: Find the derivative of the function. X +8x + 3 y3D VX

A: Here we use the quotient rule of differentiation ddxuv =vdudx-udvdxv2

Q: b) y = cosh(2Vx+1)

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Q: Use the definition of derivative to find the tangent and normal equations of f (x) = (x – 3)2 %3D

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Q: Find the second derivative of y=x-1 at x=2

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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: