FIND THE DERIVATIVE OF INDICATED VARIABLE

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 = πr⁴P / 8ηl

Solution :

F = πr⁴P / 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 = 4x⁴ + 2x³ + 3

Solution :

y = 4x⁴ + 2x³ + 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:

d⁴y/dx⁴ = 96(1) + 0

d⁴y/dx⁴ = 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 :

d⁴k/dx⁴ = 0

Problem 15 :

b = -m^-2 + 3m³

Solution :

b = -m^-2 + 3m³

First derivative:

db/dm = -(-2m^(-2-1)) + 3(3m²)

db/dm = 2m^-3 + 9m²

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:

d⁴b/dm⁴ = 24(-5m^-6 + 0)d⁴b/dm⁴ = 120m^-6