Question #306092

a) Find ๐‘‘๐‘ฆ


๐‘‘๐‘ฅ


๐‘”๐‘–๐‘ฃ๐‘’๐‘› ๐‘กโ„Ž๐‘Ž๐‘ก ๐‘ฆ = (๐‘ฅ


2 โ€“ 3x + 1)4


b) Differentiate 2


๐‘ฅ(x)โˆ’3๐‘ฅ+1



c) Intergrate 2x2 2

1
Expert's answer
2022-03-07T13:26:01-0500

a) y=(x2โˆ’3x+1)4y=(x^2-3x+1)4

we start by simplifying the equation

y=4x2โˆ’12x+4y=4x^2-12x+4

we apply the sum/difference rule

dy/dx=dy/dx = d/dx(4x2)โˆ’d/dx(12x)+d/dx(4)d/dx(4x^2)-d/dx(12x)+d/dx(4)

=8xโˆ’12=8x-12

b. y=2x(x)โˆ’3(x)+1y= 2x(x) -3(x)+1

we start by simplifying the equation

y=2x2โˆ’3x+1y=2x^2-3x+1

to find the derivative we use sum difference rule

dy/dx=d/dx(2x2)โˆ’d/dx(3x)+d/dx(1)dy/dx= d/dx(2x^2)-d/dx(3x)+d/dx(1)

=4xโˆ’3=4x-3

c. Incase the given equation is as follows

โˆซ(2x2+2)dx\int( 2x^2+2)dx

we apply the sum rule of integration

=โˆซ2x2dx+โˆซ2dx=\int2x^2dx+\int2dx

=2x3/3+2x+C=2x^3/3+2x+C

where C is the constant of integration.

Incase the given equation is as follows

= โˆซ(2x2โˆ’2)dx\int (2x^2-2)dx

we apply the difference rule of integration

=โˆซ2x2dxโˆ’โˆซ2dx=\int2x^2dx-\int 2dx

where C is the constant of integration.


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