Basic Rules of Differentiation
What is the derivative of the following function? (Use complete nomenclature).
y=x^2+x+8
y'=2x+1
What is the derivative of the following function? (As simplified as possible. Hint: No squared values)
f(x) = sin(x)tan(x)
f'(x) = tan(x)sec(x) + sin(x)
What is the derivative of the following function? (As simplified as possible).
r'(t) = 100/(e^t)
r'(t) = -100/(e^t)
What is the derivative of the following function? (Simplify as much as possible)
F(x) = tan(sin(x))
F'(x) = cos(x)sec^2(sin(x))
F'(x) = sec^2(sin(x))cos(x)
What is the derivative of the following function? (Simplify as much as possible)
y = ln(cos(x))
y' = -tan(x)
What is the derivative of the following function? (Simplify as much as possible)
q = (2r - r^2)^(1/3)
q' = (2-2r)/(3(2r-r^2)^(2/3))
What is the derivative of the following function? (Simplify as much as possible)
y^2-xy-12=0
y'=y/(2y-x)
What is the derivative of the following function? (Use complete nomenclature).
w = 3z^7 - 7z^3 + 21z^2
w' = 21z^6 - 21z^2 + 42z
What is the derivative of the following function? (As simplified as possible. Hint: Under one same denominator; common factor)
y = sqrtx*sec(x) + 3
y' = (sec(x)*(2xtan(x)+1))/(2sqrtx)
What is the derivative of the following function? (As simplified as possible.)
y = 5x^2 e^x
y' = 5x^2 e^x + 10x e^x
What is the derivative of the following function? (Simplify as much as possible)
f(x) = cos^4(7x^3 + 6x - 1)
f'(x) = (-84x^2 - 24)cos^3(7x^3 + 6x - 1)sin(7x^3 + 6x - 1)
What is the derivative of the following function? (Simplify as much as possible. Hint: Use identities to simplify)
f(ϴ) = ln(sec(ϴ) + tan(ϴ))
f'(ϴ) = sec(ϴ)
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor)
r = (csc(ϴ) + cot(ϴ))^(-1)
r' = csc(ϴ)/(csc(ϴ) + cot(ϴ))
What is the derivative of the following function? (Simplify as much as possible)
y^4=y^2-x^2
y'=(2x)/(2y-4y^3)
What is the derivative of the following function? (Use complete nomenclature).
y = (x^3)/3 + (x^2)/2 + x/4
y' = x^2 + x + 1/4
What is the derivative of the following function? (As simplified as possible. Hint: Use trig identities)
y = 4/cos(x)*sinx + 1/tan(x)
y' = 4sec^2(x) - csc^2(x)
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor)
y = 3^x x^3
y' = x^2 3^x(3 + xln(3))
What is the derivative of the following function? (Simplify as much as possible. Hint: Use identities to simplify)
y = (1-cos^2(3x))/(1 - sin^2(3x))
y' = 6sec^2(3x)tan(3x)
What is the derivative of the following function? (Simplify as much as possible).
g(x) = ln(ln(x^(1/2)))
g'(x) = 1/(2xln(sqrtx))or
g'(x)=1/(xlnx)
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor)
p = cot(sin(t)/(e^t))
p' = -((cos(t)-sin(t))/(e^t))csc^2(sin(t)/(e^t))
What is the derivative of the following function? (Simplify as much as possible)
y^2+cos(xy)-4x=5
y'=(4+ysin(xy))/(2y-xsin(xy))
What is the derivative of the following function? (Use complete nomenclature).
y = 4 - 2x - x^-3
y' = 3/(x^4) - 2
What is the derivative of the following function? (As simplified as possible. Hint: Common factor; use identities to simplify)
s = sin(t)/(1-cos(t))
s' = -1/(1-cos(t))
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor)
y = (e^x)/(π^x)
y' = -(e^x(ln(π)-1))/(π^x)
What is the derivative of the following function? (Simplify as much as possible. Hint: Use identities to simplify)
y = sec^2(x^2) + tan^2(2x)
y' = 8x sec^2(x^2) tan(x^2)
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor)
g(x) = log_4 (2x^4)
CLUE: d/dx(log_b(x))=1/(ln(b)*x)
g'(x) = 4/(xln4)
What is the derivative of the following function? Simplify as much as possible. Hint: Common factor)
r = sin(ϴ^2) cos(2ϴ)
r' = -2(sin(ϴ^2)sin(2ϴ) - ϴcos(2ϴ)cos(ϴ^2))
The slope of the tangent line to the graph of x^3-2y^3+xy=0 at the point (1,1) is:
m=4/5
What is the derivative of the following function? (Use complete nomenclature).
r = 12/ϴ - 4/(ϴ^3) + 1/(ϴ^4)
r' = -12/(ϴ^2) + 12/(ϴ^4) - 4/(ϴ^5)
What is the derivative of the following function? (Simplify as much as possible. Hint: Common factor; use identities to simplify)
f(x) = (x^3)[sin(x)cos(x)]
f'(x) = - x^2 (xsin^2(x) - xcos^2(x) - 3sin(x)cos(x))
f'(x) = x^2 (xcos(2x) - 3sin(x)cos(x))
What is the derivative of the following function? (Simplify as much as possible. Hint: leave one denominator. Common factor)
f(x) = ln(x)/(e^x)
f'(x) = (1 - xln(x))/(xe^x)
What is the derivative of the following function? (Simplify as much as possible. Hint: Use identities to simplify)
f(x) = (sin^2(5x))/(cot^2(5x)+1)
f'(x) = 20 sin^3(5x)cos(5x)
What is the derivative of the following function? (Simplify as much as possible. Hint: Use identities to simplify)
f(ϴ) = log((1+sin(ϴ))/(1-sin(ϴ)))
f'(ϴ) = (2sec(ϴ))/ln(10)
What is the derivative of the following function? (Simplify as much as possible. Hint: Leave one denominator; common factor)
f(x) = (ln(x) + xsec(x))^(1/2)
f'(x) = (1 + x^2 sec(x)tan(x) + xsec(x))/((2x(ln(x) + xsec(x))^(1/2))
Find the equation of the tangent line to the function (x+y)^3=x^3+y^3 at (-1,1).
y=-x