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)

Transformation properties
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) $$r(x) = f(-3x)$$

2021-01-23
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.