Question

The table shows some values of the derivative of an unknown function f.Complete the table by finding the derivative of each transformation of f, it possible
a) \(g(x) = f(x)\ -\ 2\)
b) \(h(x) = 2 f(x)\)
c) \(r(x) = f(-3x)\)

Answer 1

Step 1
The derivative properties:
\(\frac{d}{dx}(f(x)\ -\ a)=f'(x)\)
\(\frac{d}{dx}(af(x))=af'(x)\)
calculate the derivative of \(g(x) = f(x)\ -\ 2\) with respect to x as follows
\(g'(x)=\frac{d}{dx}(f(x)\ -\ 2)\)
\(= f'(x)\)

Step 2
Now calculate the derivative of \(h(x) = 2f(x)\) with respect to x as follows
\(h'(x)=\frac{d}{dx}=(2f(x))\)
\(= 2f'(x)\)
\(\therefore h'(x) = 2 f'(x)\)

Step 3
Now calculate the derivative of \(r(x) = f(-3x)\) with respect to x as follows
\(r'(x)=\frac{d}{dx}=f(-3x)\)
\(=\ - 3f'(-3x)\)
As \(r'(x) =\ -3f'(-3x),\) compute the the value of
\(-3f'(-3x)\)
\(r'(-2) = 3f' [-3\ (2)]\)
\(= -3 f'(6)\)
Here \(r'(-2) =\ -3f'(6)\) cannot be computed the values
of \(f'(6)\) is not known.
\(r'(-1) =\ -3f' [-3\ (-1)]\)
\(=\ -3f' (3)\)
\(=\ -3 (-5)\)
\(= 15\)
\(r'(0) =\ -3f'[-3\ (0)]\)
\(= -3(-\frac{1}{3})\)
\(= 1\)
And \(r'(1) =\ -3f[-3\ (1)]\)
\(=\ -3 f' (-3)\)
Hense \(-3 f'(-3)\) can not be compluted.