\(\int { 4xdx } =4\frac { { x }^{ 1+1 } }{ 1+1 } +c\)
\(=4\frac { { x }^{ 2 } }{ 2 } +c=2{ x }^{ 2 }+c\)

\(\int { ({ x }^{ 2 } } +x-1)dx=\int { { x }^{ 2 } } dx+\int { xdx } -\int { dx } \)
\(=\frac { { x }^{ 2+1 } }{ 2+1 } +\frac { { x }^{ 1+1 } }{ 1+1 } -x+c\)
\(=\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 2 } }{ 2 } -x+c\)
 

\(\int { 4{x }^{ 3 } } dx=4\int { { x }^{ 3 } } dx\)
\(=4\frac { { x }^{ 3+1 } }{ 3+1 } +c=4\frac { { x }^{ 4 } }{ 4 } +c={ x }^{ 4 }+c\)

Marginal cost is an addition to the total cost caused by producing one more unit of output.
MC= \(\frac { d\left( TC \right) }{ dQ } \) (or) MC= \(\frac { \triangle \left( TC \right) }{ \triangle\ Q } \)

The fundamental operation of calculus is differention dersrative is used to express the rate of change in any function.

Q_s = f (P_x) Where 'Q_s' stands for Quantity supplied of a commodity and P_x is the price of that commodity:

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(a) Computer
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TR = 20Q-2Q²     ∴ Q⁰=1
MR = 20(1)Q^1-4 - 2(2)Q^2-1
20Q^o-4Q¹
20(1) - 4Q
20-4Q
WhenQ = 3
MR = 20-4Q
20 - 4(3)
20-12
MR = 8

TC= 30+6Q²+14Q³
\(MC={d(30)\over dQ}+{d(6Q^2)\over dQ}+{d(14Q^3)\over dQ}\)
=0+ 6(2)Q^2-1 + 14(3) Q^3-1
IMC = 12Q+42Q²

Y = 2x⁴ - 6x²
Differentiate 'Y' with respect to x
\({dy\over dx}=2(4) x^{4-1}- 6(2)x^{2-1}\)
\({dy\over dx}=8x^3-12x\)

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Elasticity of supply is the relative change in supply in response to a relative change in price. If S is supply, the supply function is S = f(p)
The formula for elasticity of supply \(e_{\mathrm{s}}=\frac{\mathrm{P}}{\mathrm{S}} \cdot \frac{\mathrm{ds}}{\mathrm{dp}} ; \frac{\mathrm{ds}}{\mathrm{dp}}=\) slope

\({dq\over dp}=-4-8p\)
R=pq = p(30 - 4p - 4p²)
= 30p - 4p² - 4p³
\(MR={dR\over dq}=30{dp\over dq}-8p{dp\over dq}-12p^2{dp\over dq}\)
\(={1\over -4-8p}(-0-8p-12p^2)\)
\(={1\over 4+8p}(12p^2+8p-30)\)

Ans=\({6p^2+4p-15\over 2+4p}\)