This equation shows how to find the derivative of a polynomial term. It is often used in conjunction with the Addition Rule for Differentiation to take the derivative of a polynomial.
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( k \) | This symbol represents any given integer, \( k \in \htmlClass{sdt-0000000122}{\mathbb{Z}}\). |
| \( \:d \) | This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to. |
| \( a \) | This is a symbol for any generic constant. It can hold any numerical value |
Consider the definition of a derivative:
\[\htmlClass{sdt-0000000065}{f'(x)} = \htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000067}{h} \to 0} \frac{\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x} + \htmlClass{sdt-0000000067}{h}) - \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x})}{\htmlClass{sdt-0000000067}{h}}\]
More coming soon...
Consider \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) = \htmlClass{sdt-0000000003}{x}^2\), what is \( \htmlClass{sdt-0000000065}{f'(x)} \)?
\(\htmlClass{sdt-0000000121}{a} = 1\)
\(\htmlClass{sdt-0000000015}{k} = 2\)
We can now plug these values in to get:
\[ \htmlClass{sdt-0000000065}{f'(x)} = (2)(1)\htmlClass{sdt-0000000003}{x}^{2 - 1} = 2\htmlClass{sdt-0000000003}{x} \]