Engineering Mathematics Questions and Answers – Limits and Derivatives of Several Variables – 4

This set of Engineering Mathematics online quiz focuses on “Limits and Derivatives of Several Variables – 4”.

1. \(\frac{d(uvw)}{dx}\) is where u ,v, w are the functions of x
a) u’vw + uv’w + uvw’
b) uvw + uv’w’ + u’v’w’
c) u’v’w + uv’w’ + u’vw’
d) uv’w’ + u’v’w’ + uvw
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2. \(\frac{d(\frac{u}{v})}{dx}\) is where u, v are the functions of x
a) ^{v’u’ – uv} ⁄ _v²
b) ^{vu’ – uv’} ⁄ _v²
c) ^{vu – u’v’} ⁄ _v²
d) 0
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3. If \(y=\frac{sin(x)e^x}{cos^2(x)}\), find ^dy⁄_dx .
a) Sec² (x) e^x [1 + Tan(x)] + e^x Tan(x)Sec(x)
b) Sec² (x) e^x [Sec(x) + Tan(x)] + e^x Tan(x)Sec(x)
c) Sec² (x) e^2x [Sec(x) + Tan(x)] + e^x Tan(x)Sec(x)
d) Sec(x) e^x [Sec(x) + Tan(x)] + e^x Tan(x)Sec(x)
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4. Value of ^d⁄_dx⁡ [(1 + xe^x}{1-Cos(x))].
a) \(\frac{(1-Sin(x))(1+x) e^x + Cos(x)(1+xe^x)}{[1-Cos (x)]^2}\)
b) \(\frac{(1-Cos(x))(1+x) e^x + Sin(x)(1+xe^x)}{[1-Cos (x)]^4}\)
c) \(\frac{(1-Cos(x))(1+x) e^x + Sin(x)(1+xe^x)}{[1-Cos (x)]^2}\)
d) \(\frac{(1-Cos(x))(1+x) e^x – Sin(x)(1+xe^x)}{[1-Cos (x)]^2}\)
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5. Find the derivative of Sin(x)Tan(x) w.r.t e^x Tan(x)
a) \(\frac{Sin(x)(1+Sec^4 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)
b) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^4 (x)+Tan(x))}\)
c) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (1+Tan^2 (x)+Tan(x))}\)
d) \(\frac{Sin(x)(1+Sec^2 (x))}{e^x (2+Tan^2 (x)+Tan(x))}\)
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6. Evaluate \(\frac{d[Tan^n (x)+Tanx^n+Tan^{-1} x+Tan(nx)}{dx}]\) is
a) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nTan(nx)Sec^2 (nx)\)
b) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nSec^2 (nx)\)
c) \(nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1-x^2)+nSec^2 (nx)\)
d) \(2nTan^{n-1} xSec^2 x+nx^{n-1} Sec^2 x^n+1/(1+x^2)+nSec^2 (nx)\)
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7. Evaluate d/dx x^x ln(x)
a) x^(x-1) + x^2x ln⁡(x) + x^x [ln⁡(x)]²
b) x^(x-1) + x^x ln⁡(x) + x^x [ln⁡(x)]²
c) x^(x-1) + x^x ln⁡(x) + x^x ln⁡(x)
d) x^x + x^x ln⁡(x) + x^x [ln⁡(x)]²
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8. Evaluate the differentiation of \(tan^{-1}\frac{cos(x)-sin(x)}{cos(x)+sin(x)}\)
a) tan^-1⁡x
b) 1
c) 0
d) -1
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9. If y = Tan(x)^Tan(x) then ^dy⁄_dx⁡ = ?
a) Tan(x) [1 + lnTan(x)] Tan(x)^Tan(x)
b) Tan² (x) [1 + lnTan(x)] Tan(x)^Tan(x)
c) Sec² (x) [1 + lnTan(x)] Tan(x)^Tan(x)
d) Sec(x) [1 + lnTan(x)] Tan(x)^Tan(x)
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10. Evaluate ^d⁄_dx Cot(x)Cosec(x)
a) -Cosec² (x) – Cosec² (x)Cot(x)
b) -Cosec³ (x) – Cosec² (x)Cot(x)
c) -Cosec(x) – Cosec² (x)Cot(x)
d) -Cosec³ (x) – Cosec(x)Cot² (x)
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11. Evaluate differentiation of x² Sin(x) w.r.t Tan(x)Cosec(x)
a) \(\frac{[2xSin(x)+x^2 Cos(x)]}{-Cosec(x)-Sec^2 (x)Cosec(x)}\)
b) \(\frac{[2xSin(x)+x^2 Cos(x)]}{-Cosec(x)+Cos(x)Sin(x)}\)
c) \(\frac{[2xSin(x)+x^2 Cos(x)]}{-Cosec(x)+Sec^2 (x)Cosec(x)}\)
d) \(\frac{[2xSin(x)+x^2 Cos(x)]}{+Cosec(x)+Sec^2 (x)Cosec(x)}\)
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12. If z = e^x Sin(Cos(x))Cos(Sin(x)) Then find ^dz⁄_dx
a) [e^xSin(Cos(x))Cos(Sin(x))-e^xCos(x)Cos(Cos(x))Cos(Sin(x))-e^xSin(x)Sin(Cos(x))Sin(Sin(x))]
b) [e^xSin(Cos(x))Cos(Sin(x))-e^xSin(x)Cos(Cos(x))Cos(Sin(x))-e^xCos(x)Sin(Cos(x))Sin(Sin(x))]
c) [e^xCos(Cos(x))Sin(Sin(x))-e^xSin(x)Cos(Cos(x))Cos(Sin(x))-e^xCos(x)Sin(Cos(x))Sin(Sin(x))]
d) [e^xSin(Cos(x))Cos(Sin(x))-e^xCos(x)Cos(Cos(x))Cos(Sin(x))-e^xSin(x)Sin(Cos(x))Sin(Sin(x))]
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13. If F(x) = f(x)g(x)h(x) and F’(x) = 10F(x) and f’(x) = 10f(x) , g’(x) = 10g(x) and h’(x) = 10kh(x), then find value of k.
a) 0
b) 1
c) -1
d) 2
View Answer

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