Question

If f is a differentiable function, find an expression for the derivative of each of the following functions

Answer 1

12a. Answer:  d) x⁶ f'(x) + 6x⁵ f(x)

Step-by-step explanation:

Use the multiplication formula for derivatives:

y = a · b       →     y' = a'b + ab'

y = x⁶ · f(x)      

a = x⁶          b = f(x)

a' = 6x⁵       b' = f'(x)

y' = a'b + ab'

y' = 6x⁵ f(x) + x⁶ f'(x)

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12b. Answer:  $\bold{b)\ y=\dfrac{xf'(x)-9f(x)}{x^{10}}'}$

Step-by-step explanation:

Use the division formula for derivatives:

$y=\dfrac{a}{b}$       →     $y' = \dfrac{a'b - ab'}{b^2}$

$y=\dfrac{f(x)}{x^9}\\\\a=f(x)\qquad b=x^9\\\\a'=f(x)\qquad b'=9x^8\\\\y'=\dfrac{a'b-ab'}{b^2}\\\\y'=\dfrac{x^9f'(x)-9x^8f(x)}{(x^9)^2}\\\\.\ =\dfrac{x^9f'(x)-9x^8f(x)}{x^{18}}\\\\\text{factor out }x^{8}: y'=\dfrac{xf'(x)-9f(x)}{x^{10}}$

Note: You can also move the denominator to the top (it will have a negative exponent) and use the multiplication formula for derivatives.