The aim of this question is to learn how to evaluate the values of a higher order derivative without explicitly declaring the function itself.
![Derivative Derivative](https://www.storyofmathematics.com/wp-content/uploads/2022/11/derivative-1.png)
Derivative
To solve such problems, we may need to solve the basic rules of finding the derivatives. These include the power rule and product rule etc.
![Power of derivative Power of derivative](https://www.storyofmathematics.com/wp-content/uploads/2022/11/power-of-derivative.png)
Power of derivative
According to the power rule of differentiation:
\[ \dfrac{ d }{ dx } \bigg ( x^{ n } \bigg ) \ = \ n \ x^{ n – 1 } \]
![Product of derivative Product of derivative](https://www.storyofmathematics.com/wp-content/uploads/2022/11/product-of-derivative.png)
Product of derivative
According to the product rule of differentiation:
\[ \dfrac{ d }{ dx } \bigg ( f ( x ) \ g ( x ) \bigg ) \ = \ f^{‘} (x) \ g ( x ) \ + \ f ( x ) \ g^{‘} ( x ) \]
Expert Answer
Given:
\[ f^{‘} ( x ) \ = \ x^2 \ f ( x ) \]
Substitute $ x \ = \ 2 $ in the above equation:
\[ f^{‘} ( 2 ) \ = \ ( 2 )^{ 2 } f ( 2 ) \]
\[ f^{‘} ( 2 ) \ = \ 4 \ f ( 2 ) \]
Substitute $ f(2) \ = \ 10 $ in the above equation:
\[ f^{‘} ( 2 ) \ = \ 4 \ ( 10 ) \]
\[ f^{‘} ( 2 ) \ = \ 40 \]
Recall the given equation again:
\[ f^{‘} ( x ) \ = \ x^2 \ f ( x ) \]
Differentiating the above equation:
\[ \dfrac{ d }{ dx } \bigg ( f^{‘} ( x ) \bigg ) \ = \ \dfrac{ d }{ dx } \bigg ( x^{ 2 } f ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ \dfrac{ d }{ dx } \bigg ( x^{ 2 } \bigg ) \ f ( x ) \ + \ x^{ 2 } \ \dfrac{ d }{ dx } \bigg ( f ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ \bigg ( 2 x \bigg ) \ f(x) \ + \ x^{ 2 } \ \bigg ( f^{‘} ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ 2 x \ f(x) \ + \ x^{ 2 } \ f^{‘} ( x ) \]
Substitute $ x \ = \ 2 $ in the above equation:
\[ f^{ ” } ( 2 ) \ = \ 2 (2) \ f(2) \ + \ ( 2 )^{ 2 } f^{‘} ( 2 ) \]
\[ f^{ ” } ( 2 ) \ = \ 4 f ( 2 ) \ + \ 4 f^{‘} ( 2 ) \]
Substitute $ f ( 2 ) \ = \ 10 $ and $ f^{‘} ( 2 ) \ = \ 40 $ in the above equation:
\[ f^{ ” } ( 2 ) \ = \ 4 (10) \ + \ 4 (40) \]
\[ f^{ ” } ( 2 ) \ = \ 40 \ + \ 160 \]
\[ f^{ ” } ( 2 ) \ = \ 200 \]
Numerical Result
\[ f^{ ” } ( 2 ) \ = \ 200 \]
Example
Given that $ f ( 10 ) \ = \ 1 $ and $ f^{‘} ( x ) \ = \ x f ( x ) $, find the value of f^{ ” } ( 10 ) $.
Given:
\[ f^{‘} ( x ) \ = \ x \ f ( x ) \]
Substitute $ x \ = \ 10 $ in the above equation:
\[ f^{‘} ( 10 ) \ = \ ( 10 ) f ( 10 ) \]
Substitute $ f(10) \ = \ 1 $ in the above equation:
\[ f^{‘} ( 10 ) \ = \ 10 \ ( 1 ) \]
\[ f^{‘} ( 10 ) \ = \ 10 \]
Recall the given equation again:
\[ f^{‘} ( x ) \ = \ x \ f ( x ) \]
Differentiating the above equation:
\[ \dfrac{ d }{ dx } \bigg ( f^{‘} ( x ) \bigg ) \ = \ \dfrac{ d }{ dx } \bigg ( x f ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ \dfrac{ d }{ dx } \bigg ( x \bigg ) \ f ( x ) \ + \ x \ \dfrac{ d }{ dx } \bigg ( f ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ \bigg ( 1 \bigg ) \ f(x) \ + \ x \ \bigg ( f^{‘} ( x ) \bigg ) \]
\[ f^{ ” } ( x ) \ = \ f(x) \ + \ x \ f^{‘} ( x ) \]
Substitute $ x \ = \ 10 $ in the above equation:
\[ f^{ ” } ( 10 ) \ = \ f(10) \ + \ ( 10 ) f^{‘} ( 10 ) \]
Substitute $ f ( 10 ) \ = \ 1 $ and $ f^{‘} ( 10 ) \ = \ 10 $ in the above equation:
\[ f^{ ” } ( 10 ) \ = \ (1) \ + \ 10 (10) \]
\[ f^{ ” } ( 10 ) \ = \ 1 \ + \ 100 \]
\[ f^{ ” } ( 10 ) \ = \ 101 \]