Derivative Calculator Steps

Derivative Calculator (Power Rule)

Use this calculator to find the derivative of a single term in the form axn using the power rule.

Understanding Derivatives and the Power Rule

A derivative represents the instantaneous rate of change of a function with respect to one of its variables. Geometrically, it gives the slope of the tangent line to the graph of the function at any given point. Derivatives are fundamental in calculus and have wide applications in physics, engineering, economics, and more, for understanding rates, optimization, and motion.

The Power Rule Explained

One of the most basic and frequently used rules for differentiation is the Power Rule. It applies to functions of the form f(x) = axn, where 'a' is a constant coefficient and 'n' is any real number exponent.

The Power Rule states that the derivative of axn with respect to x is:

d/dx(axn) = anxn-1

Let's break down what this means:

1. Multiply the coefficient by the exponent: The new coefficient of the term becomes the original coefficient ('a') multiplied by the original exponent ('n').
2. Subtract one from the exponent: The new exponent of the variable ('x') becomes the original exponent ('n') minus one.

Special Cases of the Power Rule:

• Derivative of a Constant: If n=0, then f(x) = ax0 = a (a constant). Applying the power rule: a × 0 × x0-1 = 0. The derivative of any constant is always zero. This makes sense, as a constant function has no rate of change.
• Derivative of a Linear Term: If n=1, then f(x) = ax1 = ax. Applying the power rule: a × 1 × x1-1 = ax0 = a × 1 = a. The derivative of ax is simply a. This also makes sense, as the slope of a line y=ax+b is a.

How This Calculator Works

This calculator applies the power rule to a single term you provide. You input the coefficient ('a') and the exponent ('n'), and it will output the derivative of that term, along with the steps involved in applying the power rule.

Examples:

• Example 1: Find the derivative of 3x2.
  • Coefficient (a) = 3, Exponent (n) = 2
  • New Coefficient = 3 × 2 = 6
  • New Exponent = 2 - 1 = 1
  • Derivative = 6x1 = 6x
• Example 2: Find the derivative of 5x4.
  • Coefficient (a) = 5, Exponent (n) = 4
  • New Coefficient = 5 × 4 = 20
  • New Exponent = 4 - 1 = 3
  • Derivative = 20x3
• Example 3: Find the derivative of 7x.
  • Coefficient (a) = 7, Exponent (n) = 1
  • New Coefficient = 7 × 1 = 7
  • New Exponent = 1 - 1 = 0
  • Derivative = 7x0 = 7 × 1 = 7
• Example 4: Find the derivative of 10.
  • Coefficient (a) = 10, Exponent (n) = 0 (since 10 = 10x0)
  • New Coefficient = 10 × 0 = 0
  • New Exponent = 0 - 1 = -1
  • Derivative = 0x-1 = 0

Step 1: Multiply the original coefficient by the original exponent.

Step 2: Subtract 1 from the original exponent.

Result:

Result:

0

Result:

1

" + newExponent + "

Result:

" + newExponent + "

Derivative Result:

" + a + "x^{" + n + "}

Steps Applied (Power Rule):