Free Printable Partial and Total Derivatives

1.

For y=f(x, t), $\frac{\partial y}{\partial x}$ means:

a)

Differentiation of y with respect to t keeping x constant

b)

Differentiation of y with respect to both x and t

c)

Differentiation of y with respect to x keeping t constant

d)

None of these

2.

The total derivative of a function f(x, t) with respect to t is:

a)

$\frac{\text{d}f}{\text{d}t}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}.\frac{\text{d}x}{\text{d}t}$

b)

$\frac{\text{d}f}{\text{d}t}=\frac{\partial f}{\partial t}$

c)

$\frac{\text{d}f}{\text{d}t}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial x}.\frac{\text{d}x}{\text{d}t}$

d)

$\frac{\text{d}f}{\text{d}t}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x}$

3.

In signal processing, if f(x,t) represents a wave f(x,t)=Acos⁡(kx−ωt), then $\frac{\partial f}{\partial t}$ is:

a)

−Aωcos⁡(kx−ωt)

b)

−ωsin⁡(kx−ωt)

c)

−Aωsin⁡(kx−ωt)

d)

Aωsin⁡(kx−ωt)

4.

A particle moves such that its temperature is T(x,t)=x²+t², where x=t². Then $\frac{\text{d}T}{\text{d}t}$ is:

a)

4t³

b)

2t

c)

4t³+2t

d)

4t

5.

In the wave signal f(x, t)=Acos⁡(kx−ωt), the spatial derivative $\frac{\partial f}{\partial x}$ is:

a)

−Akcos⁡(kx−ωt)

b)

−Awsin⁡(kx−ωt)

c)

Aksin⁡(kx−ωt)

d)

−Aksin⁡(kx−ωt)

6.

If f(x, t)=x+t, then $\frac{\partial f}{\partial x}$ is:

a)

x

b)

1

c)

t

d)

0

7.

If f(x, t)=xt², then $\frac{\partial f}{\partial t}$ is:

a)

$2xt$

b)

2t

c)

t²

d)

x

8.

If f(x, t)=e^xt, then $\frac{\partial f}{\partial x}$ is:

a)

xe^xt

b)

e^t

c)

te^xt

d)

e^xt

9.

For f(x, t)=sin⁡(xt), the partial derivative $\frac{\partial f}{\partial t}$ is:

a)

xcos(xt)

b)

cos(xt)

c)

sinx

d)

cost

10.

In signal analysis, partial derivatives are used when:

a)

The signal is periodic

b)

The signal depends on both time and space variables

c)

The signal depends only on time

d)

The signal is constant

Partial and Total Derivatives

Total questions: 10

Worksheet time: 5mins

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