2
Differentiate y = (ax)2 + bx + c, where a, b, c are constants
dy
=
Answer: dx
Differentiate V = ln( xM − N), where M and N are constants
Answer: dV
dx =
Correct Answers:
Correct Answers:
• (2/M)*x*(xˆ(2)/M - N)ˆ(-1)
• 2*aˆ2*x+b
Differentiate y = xe(bx) , where b is a constant
dy
Answer: dx
=
Differentiate L(t) = E + eQt , where E and Q are constants
Answer: L0 (t) =
Correct Answers:
Correct Answers:
• Q*eˆ(Q*t)
• x*b*eˆ(b*x) + eˆ(b*x)
Differentiate V = Q tan(Ny) − M, where M, N and Q are constants
Answer: dV
dy =
Differentiate y = A sin(Bx +C), where A, B,C are constants
dy
=
Answer: dx
Correct Answers:
Correct Answers:
• N*Q/(cos(N*y))ˆ(2)
• A*B*cos(B*x+C)
Differentiate P(t) = Qe(kt) , where Q, k are constants
Answer: P0 (t) =
√
Differentiate z = a sx, where a and s are constants
dz
=
Answer: dx
Correct Answers:
Correct Answers:
• (a*(0.5*s))*(s*x)ˆ(-0.5)
• Q*k*eˆ(k*t)
Differentiate P(t) = Q(1.07)t , where Q is a constant
Answer: P0 (t) =
Differentiate C(y) = NeCy+S , where C, N and S are constants
Answer: C0 (y) =
Correct Answers:
Correct Answers:
• (N*C)*eˆ(C*y + S)
• Q*ln(1.07)*(1.07)ˆt
Differentiate T (t) = 200 − ae(−bt) , where a, b are constants
Answer: T 0 (t) =
p
Differentiate D(p) = 4 Ap4 − Bp −C, where A, B and C are
constants
Answer: D0 (p) =
Correct Answers:
• a*b*eˆ(-b*t)
Correct Answers:
• 2*(4*A*pˆ3 - B)/(A*pˆ4 - B*p - C)ˆ(1/2)
√
Differentiate y = x2 + a2 , where a is a constant
dy
Answer: dx
=
Differentiate B(x) =
stants
Answer: B0 (x) =
Correct Answers:
• x/sqrt(xˆ2 + aˆ2)
√
Mx2 − N, where M and N are con-
Correct Answers:
• (0.5*(2*M))*x*(M*xˆ(2) - N)ˆ(-0.5)
Differentiate y = a + b cos (πt), where a, b are constants
Answer: dy
dt =
Correct Answers:
Differentiate y = E ln(Ks + G), where E, G and K are constants
Answer: dy
ds =
• -b*pi*sin(pi*t)
Differentiate y = (ln a)x , where a is a constant > 1
Answer: dy
d =
Correct Answers:
• (E*K)*(K*s + G)ˆ(-1)
Correct Answers:
• ln(a)ˆx*ln(ln(a))
3
Differentiate f (y) = a sin( yb − c), where a, b and c are constants
Answer: f 0 (y) =
g
Differentiate s(t) = − t 2 + vt, where g, v are constants
2
Answer: s0 (t) =
Correct Answers:
• a*3*yˆ2/b*cos(yˆ3/b - c)
Correct Answers:
• -g*t + v
Differentiate f (y) = F cos(Gy + H), where F, G and H are
constants
Answer: f 0 (y) =
Differentiate B(s) = K − ln( Cs ), where K and C are constants
Answer: B0 (s) =
Correct Answers:
Correct Answers:
• -F*G*sin(G*y + H)
• (-1)*(s)ˆ(-1)
1
√
Differentiate W = w q + c, where w and c are constants
Answer: dW
dq =
Differentiate y = (x + b) tan(x + a), where a and b are constants
dy
Answer: dx
=
Correct Answers:
Correct Answers:
• (0.5*w)*(q + c)ˆ(-0.5)
Differentiate y = eMt
Answer: dy
dt =
3 −R
• tan(x+a) + (x+b)/(cos(x + a))ˆ(2)
, where M and R are constants
Differentiate B(y) = (Ay + C)euy , where A,C and u are constants
Answer: B0 (y) =
Correct Answers:
• (3*M)*tˆ(2)*eˆ(M*tˆ(3) - R)
Differentiate L(s) = sin(As4 + Bs3 +C), where A, B and C
Correct Answers:
are
• (A)*eˆ(u*y) + (A*y + C)*u*eˆ(u*y)
constants
Answer: L0 (s) =
Differentiate V = (at 2 + c) sin(bt), where a, b and c are constants
Correct Answers:
dV
• cos(A*sˆ(4) + B*sˆ(3) + C)*(4*A*sˆ(3) + 3*B*sˆ(2)) Answer: dt =
Correct Answers:
Differentiate H = K + sin(Pz), where K and P are constants
Answer: dH
dz =
• (2*a*t)*sin(b*t) + (a*tˆ2 + c)*b*cos(b*t)
√
Differentiate V = ez az − eb , where a and b are constants
Answer: dV
dz =
Correct Answers:
• P*cos(P*z)
Correct Answers:
Differentiate H = G cos(dy + w), where G, d and w are constants
Answer: dH
dy =
• eˆz*sqrt(a*z - eˆb) + (a/2)*eˆz*(1/sqrt(a*z - eˆb))
Differentiate y(x) = v sin(ex + w), where v and w are constants
Answer: y0 (x) =
Correct Answers:
• (G*(-d))*sin(d*y + w)
Correct Answers:
Differentiate y = cos(sin(A)x2 − D), where A and D are constants
dy
=
Answer: dx
• v*eˆx*cos(eˆx + w)
Differentiate V = πax−b , where a and b are constants
Answer: dV
dx =
Correct Answers:
• -2*(sin(A))*x*sin(sin(A)*xˆ(2) - D)
Differentiate C(y) =
stants
Answer: C0 (y) =
cos( E3 y3 − C2 ),
Correct Answers:
• ln(pi)*a*piˆ(a*x - b)
where C and E are con-
Differentiate C(y) = y(5.5)By , where B is a constant
Answer: C0 (y) =
Correct Answers:
Correct Answers:
• (-1*(E))*yˆ(2)*sin((E/3)*yˆ(3) - C/2)
Differentiate y
dy
Answer: dx
=
2
= ekx −c ,
• (5.5)ˆ(B*y) + B*y*ln(5.5)*(5.5)ˆ(B*y)
where k and c are constants
Differentiate g(x) = ( 12 ) px+q , where p and q are constants
Answer: g0 (x) =
Correct Answers:
Correct Answers:
• (2*k)*x*eˆ(k*xˆ(2) - c)
• ln(1/2)*(1/2)ˆ(p*x + q)*p
Differentiate W = GeKy+E , where E, G and K are constants
Answer: dW
dy =
Differentiate y = k(sin(p))x , where k and p are constants
dy
Answer: dx
=
Correct Answers:
Correct Answers:
• (G*K)*eˆ(K*y + E)
• k*ln(sin(p))*sin(p)ˆx
Differentiate B(p) = a sin(bp + c), where a, b and c are constants
Answer: B0 (p) =
Differentiate y = (b)x − cx, where b and c are constants
dy
Answer: dx
=
Correct Answers:
Correct Answers:
• ln(b)*(b)ˆx - c
• (a*b)*cos(b*p + c)
2
Differentiate B(x) = (Ce3x − D)(2x), where C and D are constants
Answer: B0 (x) =
Differentiate y = (ax + b)(cx + d), where a, b, c and d are
constants
dy
Answer: dx
=
Correct Answers:
Correct Answers:
• 3*eˆ(3*x)*C*2*x + 2*(C*eˆ(3*x) - D)
• a*(c*x + d) + c*(a*x + b)
2
Differentiate f (x) = cos(x3 − Fx5 − 2c ), where F and c are
constants
Answer: f 0 (x) =
Differentiate y = a(x − b)(x − c)(x − d), where a, b, c and d
are constants
dy
=
Answer: dx
Correct Answers:
• (-1)*sin(xˆ(3) - (F/5)*xˆ(2) - c/2)*(3*xˆ(2) - (2*(F/5))*x)
Correct Answers:
• a*((x-c)*(x-d) + (x-b)*(x-d) + (x-b)*(x-c))
Differentiate y =
Answer:
dy
dx
a−x
, where a and b are constants
b+x
Differentiate y =
Answer:
=
dy
dx
ax − b
, where a and b are constants
a − bx
=
Correct Answers:
• (a*(a - b*x) + b*(a*x - b))/(a - b*x)ˆ2
Correct Answers:
• (-(b + x) - (a - x))/(b + x)ˆ2
Differentiate y = ax
dy
Answer: dx
=
Differentiate H = (ax + b)(cx)3 , where a, b and c are constants
Answer: dH
dx =
√
b,
where a and b are constants
Correct Answers:
• a*sqrt(b)*xˆ(sqrt(b) - 1)
Correct Answers:
Differentiate y =
dy
Answer: dx
=
• 4*a*cˆ3*xˆ3 + 3*b*cˆ3*xˆ2
Differentiate f (x) = ax ln(bx + c), where a, b and c are constants
Answer: f 0 (x) =
√ x
a b , where a and b are constants
Correct Answers:
• sqrt(a)*ln(b)*bˆx
1
Correct Answers:
Differentiate y = 3x n+2 − m, where m and n are constants
dy
Answer: dx
=
• a*ln(b*x + c) + a*b*x/(b*x + c)
Correct Answers:
• 3*(1/(n+2))*xˆ(-1+1/(n+2))
Differentiate y = (ax + b) sin(2x − 1), where a and b are constants
dy
Answer: dx
=
Differentiate y =
Correct Answers:
Answer:
• a*sin(2*x - 1) + (a*x + b)*2*cos(2*x - 1)
dy
dx
n+2
− mx, where m and n are constants
3x
=
Correct Answers:
• -1*((n+2)/3)*xˆ(-2) - m
Differentiate g(x) = (mx + b) cos( 5x ), where m and b are constants
Answer: g0 (x) =
Differentiate y =
Correct Answers:
Answer:
dy
dx
mx
− 3, where m and n are constants
n+2
=
Correct Answers:
• m/(n+2)
• m*cos(x/5) - (m*x + b)*sin(x/5)/5
Differentiate B(x) = a tan( 2x ) − bx, where a and b are constants
Answer: B0 (x) =
Differentiate y = ax1/b , where a and b are constants
dy
Answer: dx
=
Correct Answers:
• a*(1/b)*xˆ(-1+1/b)
Correct Answers:
• a/(2*cos(x/2)ˆ2) - b
Differentiate f (x) = A ln(sin(Bx)), where A and B are constants
Answer: f 0 (x) =
Differentiate W = (Cx + D)12x , where C and D are constants
Answer: dW
dx =
Correct Answers:
Correct Answers:
• A*B*cos(B*x)/sin(B*x)
• C*12ˆx + (C*x + D)*ln(12)*12ˆx
3
Differentiate g(x) = A − sin(cos(Cx)), where A and C are
constants
Answer: g0 (x) =
Differentiate y =
Answer:
Correct Answers:
dy
dx
k
, where k, p and q are constants
p − qx
=
Correct Answers:
• C*sin(C*x)*cos(cos(C*x))
Differentiate h(x) =
stant.
Answer: h0 (x) =
p
• k*q*(p - q*x)ˆ(-2)
a + (x − b)2 , where a and b are con-
Differentiate z = u sin(wx), where u and w are constants
dz
Answer: dx
=
Correct Answers:
Correct Answers:
• u*w*cos(w*x)
• (x - b)/(a + (x - b)ˆ2)ˆ(1/2)
Differentiate z =
dz
Answer: dx
=
p
Differentiate L(y) = P − (W − 2y), where P and W are constants
Answer: L0 (y) =
(ax − b)3 , where a and b are consants
Correct Answers:
Correct Answers:
• 2
• (3/2)*a*(a*x - b)ˆ(1/2)
Differentiate A(x) = 12 (P − 2x)x, where P is a constant
Answer: A0 (x) =
Differentiate P(y) = sin(ln(Py−Q)), where P and Q are constants
Answer: P0 (y) =
Correct Answers:
Correct Answers:
• P/2 - 2*x
• (P/(P*y - Q))*cos(ln(P*y - Q))
Differentiate V (x) = x(L − 2x)(W − 2x), where L and W are
constants.
Answer: V 0 (x) =
Differentiate G(x) = R sin2 (Sx), where R and S are constants
Answer: G0 (x) =
Correct Answers:
Correct Answers:
• (L - 2*x)*(W - 2*x) - 2*x*(W + L - 4*x)
• 2*R*S*sin(S*x)*cos(S*x)
√
Differentiate W = k y + A, where k and A are constants.
Answer: dW
dy =
Differentiate f (x) = P cos2 (Qx), where P and Q are constants
Answer: f 0 (x) =
Correct Answers:
Correct Answers:
• -2*P*Q*cos(Q*x)*sin(Q*x)
• (k/2)*(y + A)ˆ(-1/2)
Differentiate P(z) = A ln2 (Bz), where A and B are constants
Answer: P0 (z) =
Differentiate S(r) = 4π(r − R)2 , where R is a constant.
Answer: S0 (r) =
Correct Answers:
Correct Answers:
• 2*A*ln(B*z)/z
• 8*pi*(r - R)
Differentiate H(t) = a tan2 (kt), where a and k are constants
Answer: H 0 (t) =
Differentiate g(y) = A cos(B − ln(y)), where A and B are constants.
Answer: g0 (y) =
Correct Answers:
• 2*a*k*tan(k*t)/cos(k*t)ˆ2
Correct Answers:
• A*(1/y)*sin(B - ln(y))
q
Differentiate f (t) = ( a − bt )5 , where a and b are constants.
Answer: f 0 (t) =
Differentiate g(y) =
Answer: g0 (y) =
Correct Answers:
p
(L − y)2 + y2 , where L is a constant.
Correct Answers:
• (5/2)*(-1/b)*(a - t/b)ˆ(3/2)
• (2*y - L)/((L - y)ˆ2 + yˆ2)ˆ(1/2)
Differentiate g(x) = (kx)3 − (mx)2 , where k and m are constants
Answer: g0 (x) =
Differentiate g(y) = P cos(2Py), where P is a constant
Answer: g0 (y) =
Correct Answers:
Correct Answers:
• -2*Pˆ2*sin(2*P*y)
• 3*kˆ3*xˆ2 - 2*mˆ2*x
4
Differentiate h(r) =
Answer: h0 (r) =
2
r
+ ear , where a is a constant.
a
Differentiate y = cos(a − bπt), where a and b are constants.
Answer: dy
dt =
Correct Answers:
• b*pi*sin(a - b*pi*t)
Correct Answers:
• (1/a) + 2*a*r*eˆ(a*rˆ2)
√
Differentiate f (x) = A( 2)x , where A is a constant.
Answer: f 0 (x) =
Differentiate h(r) = ln(sin(Pr)), where P is a constant.
Answer: h0 (r) =
Correct Answers:
• A*ln(sqrt(2))*(sqrt(2))ˆx
Correct Answers:
• P*cos(P*r)/sin(P*r)
Differentiate g(t) = h − vt − 2a t 2 , where h, v and a are constants.
Answer: g0 (t) =
b
Differentiate h(r) = 2ar2 + , where a and b are constants.
r
Answer: h0 (r) =
Correct Answers:
• -v - a*t
Correct Answers:
• 4*a*r - b*rˆ(-2)
√
Differentiate h(r) = mr8 b − r2 , where m and b are constants.
Answer: h0 (r) =
Differentiate y = (a − x)(cx2 − d), where a, c and d are constants.
dy
Answer: dx
=
Correct Answers:
• -1*(c*xˆ2 - d) + (a - x)*c*2*x
Correct Answers:
• 8*m*rˆ7*sqrt(b - rˆ2) - m*rˆ9*(b - rˆ2)ˆ(-1/2)
Differentiate z = a − sin( πb x), where a and b are constants.
dz
Answer: dx
=
Differentiate D(y) = R − S(y2 − RS ), where R and S are constants.
Answer: D0 (y) =
Correct Answers:
• -(pi/b)*cos(pi*x/b)
Correct Answers:
• -2*S*y
F(t) = L − a( 12 )t ,
Differentiate
Answer: F 0 (t) =
Differentiate r = (a + bx2 )0.25 , where a and b are constants.
dr
Answer: dx
=
where L and a are constants.
Correct Answers:
• (1/4)*(a + b*xˆ2)ˆ(-3/4)*2*b*x
Correct Answers:
• -1*ln(1/2)*a*(1/2)ˆt
Differentiate Q =
Answer: dQ
dx =
Differentiate f (t) = ta−t , where a > 1 is a constant
Answer: f 0 (t) =
p
1 − c x1/4 , where c is a constant.
Correct Answers:
• (1/2)*(1 - c*xˆ(1/4))ˆ(-1/2)*(-1/4)*c*xˆ(-3/4)
Correct Answers:
• aˆ(-t) - t*ln(a)*aˆ(-t)
Differentiate G = px5 + qx3 − xr , where p, q and r are constants.
Answer: dG
dx =
√
Differentiate y = a2 − x2 , where a is a constant.
dy
Answer: dx
=
Correct Answers:
• 5*p*xˆ4 + 3*q*xˆ2 - r*xˆ(r-1)
Correct Answers:
• -1*x/(aˆ2 - xˆ2)ˆ(1/2)
Differentiate y = axr , where a and r are constants.
dy
Answer: dx
=
Differentiate y = (a − bx)3 , where a and b are constants.
dy
Answer: dx
=
Correct Answers:
• r*a*xˆ(r-1)
Correct Answers:
• -3*b*(a - b*x)ˆ2
Differentiate f (z) = ln(ln(a))z − c, where a > 0 and c are
constants.
Answer: f 0 (z) =
Differentiate G(t) = R(5.125)−t , where R is a constant.
Answer: G0 (t) =
Correct Answers:
• -1*R*ln(5.125)*(5.125)ˆ(-t)
Correct Answers:
• ln(ln(a))
c
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5