DERIVATIVE OF X WITH RESPECT TO X

Derivative measures the rate of change in one variable with respect to the change in the other variable.

For example, let x be an independent variable and y be a dependent variable.

So, the value of y is getting changed according to the change in the value of x.

Then, the derivative of y with respect to x is

dy/dx

Power Rule of Derivative

Power rule of derivative is the fundamental tool to find the derivative of a function in the form

y = xⁿ

To get the derivative of xⁿ with respect to x, we have to bring the exponent n in front of x and subtract 1 from the exponent.

dy/dx = nx^{n - 1}

Example 1 :

y = x³

dy/dx = 3x^{3 - 1}

= 3x²

Example 2 :

y = x^2.

dy/dx = 2x^{2 - 1}

= 2x¹

= 2x

Derivative of x with Respect to x

We can find the derivative of x with respect to x using the power rule of derivative.

y = x

y = x¹

Use the power rule of derivative.

dy/dx = 1x^{1 - 1}

= 1x⁰

= 1(1)

= 1

So, the derivative of x with respect to x is 1.

We can extend this concept to any variable like y or t.

Derivative of y with respect to y is 1.

Derivative of t with respect to t is 1.

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