Below are eight functions. Find the first derivative of each. Space is provided.

Question

Below are eight functions. Find the first derivative of each. Space is provided. Use good dark ink if you are returning this by a scanned version. All of the derivatives can be found by using combinations of the constant rule, power function rule and sum-difference rule. Do not use the product rule. It is not needed. The degree of difficulty (more or less) increases from (a-h). Be sure to show intermediate work. Check the scoring rubric to see how the points are awarded. For example, the first derivative of the function: 's(2x +3)2 is 2(2x 3X2) dx Although this can be simplified to: # 8X +12 ax please do not simplify your answer. I can better diagnose your differentiation difficulties. This also avoids more algebraic mistakes on your part. Hint: To make either (g) or (h) easier, multiply the two parts together first, then find the derivative of the resulting product. This avoids the need to use the product rule. Good Luck! a) E=4+63-2G2 dE/dG- b) Z 25+16x3-4x2 e) ?-ias-iss-2+55-3-12 dQ/dS- c) W X-3x +3x dW/dX- t R 25 7 +6T +11T2 dR/dT- dWdY- dQ/dV

Explanation / Answer

Answer : a) Given, E = 4 + G3 - 2G2

The "first derivative" for E is

dE/dG = d(4 + G3 - 2G2) / dG = 0 + 3*G2 - 2*2G  

=> dE/dG = 3G2 - 4G

b) Given, Z = 25 + 16X3 - 4X2

The "first derivative" for Z is

dZ/dX = d(25 + 16X3 - 4X2)/dX = 0 + 3*16X2 - 2*4X

=> dZ/dX = 48X2 - 8X  

c) W = X2/3 - 3X3 + 3X1/3

The "first derivative" for W is

dW/dX = d(X2/3 - 3X3 + 3X1/3 )/dX = 2/3*X(2/3 - 1) - 3*3X2 + 1/3*3X(1/3 - 1)

=> dW/dX = 2/3X-1/3 - 9X2 + X-2/3

d) S = X3 - 2X1/2 + 3X2/3

The "first derivative" for S is

dS/dX = d(X3 - 2X1/2 + 3X2/3)/dX = 3*X2 - 1/2*X(1/2 - 1) + 2/3*3X(2/3 - 1)

=> dS/dX = 3X2 - 1/2X-1/2 + 2X-1/3

e) Q = 10S - 15S-2 + 5S-3 - 12

The "first derivative" for Q is

dQ/dS = d(10S - 15S-2 + 5S-3 - 12)/dS = 10 - (-2)15S-2-1 + (-3)5S-3-1

=> dQ/dS= 10 + 30S-3 - 15S-4

f) R = 25 - 7T-3 + 6T-1 + 11T2

The "first derivative" for R is

dR/dT = d(25 - 7T-3 + 6T-1 + 11T2)/dT = - (-3)*7T-3-1 + (-1)*6T-1-1 + 2*11T

=> dR/dT = 21T-4 - 6T-2 + 22T

g) W = (2Y - 6Y3)(Y4) = 2Y5 - 6Y7

The "first derivative" for W is

dW/dY = d(2Y5 - 6Y7)/dY = 5*2Y5-1 - 7*6Y7-1

=> dW/dY= 10Y4 - 42Y6

h) Q = V3(3 + 2V + V2) = 3V3 + 2V4 + V5

The "first derivative" for Q is

dQ/dV = d(3V3 + 2V4 + V5)/dV = 3*3V2 + 4*2V3 + 5V4

=> dQ/dV = 9V2 + 8V3 + 5V4

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