Calculus Calculator

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Power Rule Calculus Calculator

Enter a term in the form: axn


For Definite Integrals (Optional):

Results will appear here…

Comprehensive Guide to Power Rule Calculus

Calculus is the mathematical study of continuous change. Two of its primary branches are differential calculus (concerning rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and areas under curves). This calculator focuses on the Power Rule, the most fundamental rule for both derivatives and integrals.

The Derivative Power Rule

The Power Rule for derivatives is used to find the slope of a function at any given point. If you have a function in the form f(x) = axn, the derivative is calculated as:

Formula: d/dx [axn] = n * ax(n-1)

Example: If you have 5x3, the derivative is (3 * 5)x(3-1), which simplifies to 15x2.

The Integral Power Rule

Integration is the reverse process of differentiation. For any term axn (where n ≠ -1), the indefinite integral is:

Formula: ∫ axn dx = (a / (n+1))x(n+1) + C

Note: If the exponent (n) is -1, the integral is a * ln|x| + C. Our calculator handles standard power rule calculations and definite integrals over a specific range.

Common Calculus Rules Reference Table

Function Type Derivative f'(x) Indefinite Integral ∫f(x)dx
Constant (c) 0 cx + C
xn nxn-1 (xn+1)/(n+1) + C
ex ex ex + C
ln(x) 1/x x ln(x) – x + C

Frequently Asked Questions

What is a definite integral?
A definite integral calculates the signed area between the function and the x-axis within a specific interval [a, b]. It results in a single number rather than a function.

When can I not use the Power Rule?
The Power Rule for integration cannot be used when the exponent n is -1, because division by zero would occur. In that case, the integral is the natural logarithm function.

function calculateDerivative() { var a = parseFloat(document.getElementById('coeff').value); var n = parseFloat(document.getElementById('exponent').value); var resultDiv = document.getElementById('calcResult'); if (isNaN(a) || isNaN(n)) { resultDiv.innerHTML = "Please enter valid numbers for the coefficient and exponent."; return; } var newCoeff = a * n; var newExp = n – 1; var output = "Derivative Result:"; output += "Function: f(x) = " + a + "x" + n + ""; if (n === 0) { output += "Derivative: f'(x) = 0"; } else if (newExp === 0) { output += "Derivative: f'(x) = " + newCoeff; } else if (newExp === 1) { output += "Derivative: f'(x) = " + newCoeff + "x"; } else { output += "Derivative: f'(x) = " + newCoeff + "x" + newExp + ""; } resultDiv.innerHTML = "
" + output + "
"; } function calculateIntegral() { var a = parseFloat(document.getElementById('coeff').value); var n = parseFloat(document.getElementById('exponent').value); var lower = parseFloat(document.getElementById('lowerLimit').value); var upper = parseFloat(document.getElementById('upperLimit').value); var resultDiv = document.getElementById('calcResult'); if (isNaN(a) || isNaN(n)) { resultDiv.innerHTML = "Please enter valid numbers for the coefficient and exponent."; return; } if (n === -1) { var indef = a + " ln|x| + C"; var output = "Indefinite Integral:∫ " + a + "x-1 dx = " + indef + ""; if (!isNaN(lower) && !isNaN(upper)) { if (lower <= 0 || upper <= 0) { output += "Note: For ln|x|, limits must be positive to calculate a real definite integral."; } else { var defVal = a * (Math.log(Math.abs(upper)) – Math.log(Math.abs(lower))); output += "Definite Integral [" + lower + ", " + upper + "]:"; output += "" + defVal.toFixed(4) + ""; } } resultDiv.innerHTML = output; return; } var newCoeff = a / (n + 1); var newExp = n + 1; var output = "Indefinite Integral:"; output += "∫ " + a + "x" + n + " dx = " + newCoeff.toFixed(4) + "x" + newExp + " + C"; if (!isNaN(lower) && !isNaN(upper)) { var valUpper = (a / (n + 1)) * Math.pow(upper, n + 1); var valLower = (a / (n + 1)) * Math.pow(lower, n + 1); var defResult = valUpper – valLower; output += "Definite Integral Result:"; output += "Interval: [" + lower + ", " + upper + "]"; output += "Value: " + defResult.toFixed(4) + ""; } resultDiv.innerHTML = output; }

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