# 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.