Question

If $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ , then how do you compute $ f'\left( 3 \right) $ ?

Answer 1

Hint: In the given problem, we are required to differentiate the function given to us as $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ with respect to x and then substitute the value of x as given in the problem. Since, $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ is function involving powers of a variable, so we will have to apply the power rule of differentiation in the process of differentiating $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ . Also, the power rule of differentiation must be remembered so as to solve the given problem.

Complete step-by-step answer:
To find derivative of $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ with respect to x, we have to find differentiate $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ with respect to $ x $ .
So, Derivative of $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ with respect to $ x $ can be calculated as $ \dfrac{d}{{dx}}\left( {{x^{\dfrac{{ - 5}}{7}}}} \right) $ .
Now, $ f'\left( x \right) = \dfrac{d}{{dx}}\left( {{x^{\dfrac{{ - 5}}{7}}}} \right) $
Using the power rule of differentiation $ \dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}} $ , we get ,
 $ \Rightarrow f'\left( x \right) = \left( {\dfrac{{ - 5}}{7}} \right){x^{\left( {\dfrac{{ - 12}}{7}} \right)}} $
So, we get the derivative of the given function $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ as $ f'\left( x \right) = \left( {\dfrac{{ - 5}}{7}} \right){x^{\left( {\dfrac{{ - 12}}{7}} \right)}} $ . Now, we have to put the value of variable x as $ 3 $ , we get,
 $ \Rightarrow f'\left( 3 \right) = \left( {\dfrac{{ - 5}}{7}} \right){\left( 3 \right)^{\left( {\dfrac{{ - 12}}{7}} \right)}} $
Simplifying the expression, we get,
 $ \Rightarrow f'\left( 3 \right) = \left( {\dfrac{{ - 5}}{7}} \right)\left( {\dfrac{1}{{{3^{\dfrac{{12}}{7}}}}}} \right) $
 $ \Rightarrow f'\left( 3 \right) = \left( {\dfrac{1}{3}} \right)\left( {\dfrac{{ - 5}}{7}} \right)\left( {\dfrac{1}{{{3^{\dfrac{5}{7}}}}}} \right) $
So, the derivative of $ f\left( x \right) = {x^{\dfrac{{ - 5}}{7}}} $ with respect to x for the value of variable x as $ 3 $ is
$ f'\left( 3 \right) = \left( {\dfrac{1}{3}} \right)\left( {\dfrac{{ - 5}}{7}} \right)\left( {\dfrac{1}{{{3^{\dfrac{5}{7}}}}}} \right) $ .
So, the correct answer is “$ f'\left( 3 \right) = \left( {\dfrac{1}{3}} \right)\left( {\dfrac{{ - 5}}{7}} \right)\left( {\dfrac{1}{{{3^{\dfrac{5}{7}}}}}} \right) $”.

Note: The given problem may also be solved using the first principle of differentiation.
In the given question, we are required to find the value of the first derivative of the given function for a given value of variable x. The derivatives of basic trigonometric functions must be learned by heart in order to find derivatives of complex functions. Power rule of differentiation is to be used in order to find the derivative of the function f given to us.