`f(x) = e^x - x^3` Find the first and second derivatives of the function. Check to see that your answers aer reasonable by comparing the graphs of f, f', and f''

Textbook Question

Chapter 3, 3.1 - Problem 46 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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nees101's profile pic

nees101 | Student, Graduate | (Level 2) Adjunct Educator

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Given function is:

`f(x)=e^x-x^3`

Now we will find the first derivative:

`f'(x)=d/dx(e^x-x^3)`

     `=d/dx(e^x)-d/dx(x^3)`

     `=e^x-3x^2`

The second derivative is obtained by taking the derivative of the first derivative.

`f''(x)=d/dx(f'(x))=d/dx(e^x-3x^2)`

                       `=d/dx(e^x)-d/dx(3x^2)`

                       `=e^x-6x`

Comparison of graphs of f,f' and f'' is shown below:

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a-maths-9's profile pic

a-maths-9 | Middle School Teacher | (Level 1) Adjunct Educator

Posted on

`f(x)=e^x-x^3`

`f'(x)=d/dx(e^x-x^3)`

`f'(x)=d/dx(e^x)-d/dx(x^3)`

`f'(x)=e^x-3x^2`

now we will find 2nd derivative

`f''(x)=d/dx(e^x)-d/dx(3x^2)`

`f''(X)=e^x-3*2(x)`

`f''(x)=e^x-6x`

hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

Posted on

Note:- 

1) If f(x) = e^(nx) ; where n = a real number;

then f'(x) = n*e^(nx)

2) If f(x) = x^n ; where n = a real number

then f'(x) = n*x^(n-1)

Now, given f(x) = (e^x) -  (x^3)

Thus, f'(x) = (e^x) - 3*(x^2)

f"(x) = (e^x) - 6*(x^1) 

or, f"(x) = (e^x) - 6x

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