# The differentiation of the given function

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.1, Problem 46E

(A)

To determine

## To show:The differentiation of the given function

Expert Solution

d2ydx2=12t212t+1

### Explanation of Solution

Given:

The equation of motion of a particle is

s=t42t3+t2t

Where f and g have derivatives of all orders

Concept used:

Definition of the differentiation:-Differentiation is the action of computing a derivative

The derivative of a function y=f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to x

Calculation:

The function

s=t42t3+t2t...................(1)

The derivative of a function

y=f(x)y=f(x)=dydx

Differentiating the equation (1) with respect to x

s=dydx=ddx(t42t3+t2t)s=4t36t2+t1

Again Differentiating the above equation with respect to x

d2ydx2=ddx(4t36t2+t1)d2ydx2=12t212t+1

(B)

To determine

Expert Solution

### Explanation of Solution

Given:

The equation of motion of a particle is

s=t42t3+t2t

Where f and g have derivatives of all orders

Concept used:

Definition of the differentiation:-Differentiation is the action of computing a derivative

The derivative of a function y=f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to x

Calculation:

The function

s=t42t3+t2t...................(1)

The derivative of a function

y=f(x)y=f(x)=dydx

Differentiating the equation (1) with respect to x

s=dydx=ddx(t42t3+t2t)s=4t36t2+t1

Again Differentiating the above equation with respect to x

d2ydx2=ddx(4t36t2+t1)d2ydx2=12t212t+1

Putting t=1

d2ydx2=12t212t+1s=1212+1s=1

(c)

To determine

Expert Solution

t=112,t=0

### Explanation of Solution

Given:

The equation of motion of a particle is

s=t42t3+t2t

Where f and g have derivatives of all orders

Concept used:

Definition of the differentiation:-Differentiation is the action of computing a derivative

The derivative of a function y=f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to x

Calculation:

The function

s=t42t3+t2t...................(1)

The derivative of a function

y=f(x)y=f(x)=dydx

Differentiating the equation (1) with respect to x

s=dydx=ddx(t42t3+t2t)s=4t36t2+t1

Again Differentiating the above equation with respect to x

d2ydx2=ddx(4t36t2+t1)d2ydx2=12t212t+1

Putting the s=0

s=12t212t+112t212t+1=012t212t=112t(t1)=1t=112,t=0

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