Differentiation of kx^n

Differentiation of kx^n

Basic Rule of Differentiation of $kx^n$

• The differential of a function in the form $kx^n$ is given by the expression $nkx^{n-1}$.
• Here, $n$ is the power of $x$, $k$ is a constant factor which will remain same and $x$ is the variable.

Understanding Differentiation of $kx^n$

• When the function $f(x) = kx^n$ is differentiated with respect to $x$, the power $n$ is brought down and multiplied with the existing constant $k$. The new power for $x$ becomes $n-1$.
• This rule is a direct consequence of the basic differentiation rules and Power Rule of derivatives.
• The power rule states that if $n$ is a real number and $f(x) = x^n$, then $f’(x) = nx^{n-1}$.

Examples

• Example 1: For a function $f(x) = 3x^4$, its derivative $f’(x)$ will be $4*3x^{4-1} = 12x^3$.
  • Here, $3$ is the constant $k$ and $4$ is the power $n$.
• Example 2: If we have a function $f(x) = 7x^5$, the derivative $f’(x)$ will be $5*7x^{5-1} = 35x^4$.
  • In this case, $7$ is the constant $k$ and $5$ is the power $n$.

Note

• Derivative of a function is just another word for the gradient or rate of change of the function.

• For functions in the form $kx^n$, the Power Rule for Differentiation is applied. This forms the basis for Differentiation of $kx^n$.

• In the examinations, candidates need to be quick and precise in applying this rule. Regular practise with a variety of examples is recommended.