Find the derivative with respect to the variable indicated:
Problem 1 :
Find the derivative with respect to "r".
C = 2πr
Solution :
C = 2πr
dC/dr = 2π
Problem 2 :
Find the derivative with respect to "x".
y = mx + b
Solution :
y = mx + b
dy/dx = m + 0
dy/dx = m
Problem 3 :
Find the derivative with respect to "t".
v = u + at
Solution :
v = u + at
dv/dt = 0 + a
dv/dt = a
Problem 4 :
Find the derivative with respect to "r".
A = πr²
Solution :
A = πr²
dA/dr = 2πr
Problem 5 :
Find the derivative with respect to "v".
E = 1/2 mv²
Solution :
E = 1/2 mv²
dE/dv = 1/2 (2mv)
dE/dv = mv
Problem 6 :
Find the derivative with respect to "V".
P = V²/R
Solution :
P = V²/R
dP/dV = 2V/R
Problem 7 :
s = ut + 1/2 at²
Solution :
s = ut + 1/2 at²
ds/dt = u + 1/2 (2at)
ds/dt = u + at
Problem 8 :
Find the derivative with respect to "R".
P = RI²
Solution :
P = RI²
dP/dI = 2RI
Problem 9 :
Find the derivative with respect to "r".
V = 4πr³ / 3
Solution :
V = 4πr³ / 3
dV/dr = 12πr² / 3
dV/dr = 4πr²
Problem 10 :
F = πr^{4}P / 8ηl
Solution :
F = πr^{4}P / 8ηl
dF/dr = 4πr³P / 8ηl
dF/dr = πr³P / 2ηl
Problem 11 :
T = 2π √l/g
Solution :
T = 2π √l/g
T = 2π (l/g)^{1/2}
dT/dl = 2π · 1/2(l/g)^{(1/2-1)} (l/g)
= π · 1/g (l) / (√l/g)
= π / g√l/g
dT/dl = π / √gl
Problem 12 :
v = √2GM/r
Solution :
Calculate the first, second, third, and fourth derivatives of:
Problem 13 :
y = 4x^{4} + 2x^{3} + 3
Solution :
y = 4x^{4} + 2x^{3} + 3
First derivative:
dy/dx = 4(4x³) + 2(3x²) + 0
dy/dx = 16x³ + 6x²
Second derivative:
d²y/dx² = 16(3x²) + 6(2x)
d²y/dx² = 48x² + 12x
Third derivative:
d³y/dx³ = 48(2x) + 12
d³y/dx³ = 96x + 12
Fourth derivative:
d^{4}y/dx^{4} = 96(1) + 0
d^{4}y/dx^{4} = 96
Problem 14 :
k = x³ + 7x - 11
Solution :
k = x³ + 7x - 11
First derivative :
dk/dx = 3x² + 7 - 0
dk/dx = 3x² + 7
Second derivative :
d²k/dx² = 3(2x) + 0
d²k/dx² = 6x
Third derivative :
d³k/dx³ = 6(1)
d³k/dx³ = 6
Fourth derivative :
d^{4}k/dx^{4} = 0
Problem 15 :
b = -m^{-2} + 3m^{3}
Solution :
b = -m^{-2} + 3m^{3}
First derivative:
db/dm = -(-2m^{(-2-1)}) + 3(3m²)
db/dm = 2m^{-3 }+ 9m^{2}
Second derivative:
d²b/dm² = 2(-3m^{-4}) + 9(2m)
d²b/dm² = -6m^{-4} + 18m
Third derivative:
d³b/dm³ = -6(-4m^{-5}) + 18
d³b/dm³ = 24m^{-5} + 18
Fourth derivative:
d^{4}b/dm^{4} = 24(-5m^{-6} + 0)d^{4}b/dm^{4} = 120m^{-6}
Dec 08, 23 08:03 AM
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