Derivative Calculator

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Derivative Calculator
Power Rule (ax^n)Sine Rule (a sin(nx))Cosine Rule (a cos(nx))Exponential Rule (a e^nx)Natural Log Rule (a ln(nx))
Result f'(x):
function updateLabels(){var type=document.getElementById('calc_type').value;var l1=document.getElementById('label1');var l2=document.getElementById('label2′);if(type==='power'){l1.innerHTML='Coefficient (a):';l2.innerHTML='Exponent (n):';}else if(type==='ln'){l1.innerHTML='Coefficient (a):';l2.innerHTML='Inner Multiplier (n):';}else{l1.innerHTML='Coefficient (a):';l2.innerHTML='Inner Value (n):';}}function calculateDerivative(){var a=parseFloat(document.getElementById('input1').value);var n=parseFloat(document.getElementById('input2').value);var type=document.getElementById('calc_type').value;var showSteps=document.getElementById('showSteps').checked;if(isNaN(a)||isNaN(n)){alert('Please enter valid numerical values.');return;}var resultText="";var stepText="";var prod=a*n;if(type==='power'){if(n===0){resultText="0″;stepText="The derivative of a constant is 0.";}else if(n===1){resultText=a.toString();stepText="Using the power rule: d/dx["+a+"x] = "+a;}else{var newExp=n-1;var expPart=newExp===1?"x":"x^"+newExp;resultText=prod+expPart;stepText="Power Rule: multiply coefficient ("+a+") by exponent ("+n+"), then subtract 1 from exponent. Result: "+prod+"x^"+newExp;}}else if(type==='sin'){resultText=prod+" cos("+n+"x)";stepText="Chain Rule: d/dx["+a+"sin("+n+"x)] = "+a+" * cos("+n+"x) * "+n+" = "+prod+"cos("+n+"x)";}else if(type==='cos'){resultText=(-1*prod)+" sin("+n+"x)";stepText="Chain Rule: d/dx["+a+"cos("+n+"x)] = "+a+" * -sin("+n+"x) * "+n+" = "+(-1*prod)+"sin("+n+"x)";}else if(type==='exp'){resultText=prod+" e^("+n+"x)";stepText="Exponential Rule: d/dx["+a+"e^("+n+"x)] = "+a+" * e^("+n+"x) * "+n+" = "+prod+"e^("+n+"x)";}else if(type==='ln'){resultText=a+" / x";stepText="Log Rule: d/dx["+a+"ln("+n+"x)] = "+a+" * (1/"+n+"x) * "+n+" = "+a+"/x. Note: n cancels out.";}document.getElementById('resultFormula').innerHTML="f'(x) = "+resultText;document.getElementById('stepDetails').innerHTML=stepText;document.getElementById('stepDetails').style.display=showSteps?'block':'none';document.getElementById('answer').style.display='block';}

How to Use the Derivative Calculator

The derivative calculator is a powerful tool designed to help students, engineers, and mathematicians find the instantaneous rate of change of a function. By entering the components of a mathematical expression, you can quickly find its first derivative without manually applying complex calculus rules.

To get started, simply select the type of function you are working with from the dropdown menu and input the required coefficients and exponents.

Choose Differentiation Rule
Select the basic structure of your function, such as Power Rule (ax^n), Sine, Cosine, or Exponential functions.
Coefficient (a)
The constant number that multiplies the variable or function (e.g., in 5x^2, the coefficient is 5).
Exponent or Inner Value (n)
The power to which the variable is raised or the multiplier inside a trigonometric/exponential function.

How It Works: Understanding the Math

When you use this derivative calculator, the script applies fundamental calculus identities. The most common rule used is the Power Rule, which states:

d/dx [ax^n] = n · ax^(n-1)

For more complex functions, the calculator utilizes the Chain Rule. For example, when differentiating a trigonometric function like sin(nx), the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

  • Sine Rule: d/dx [sin(u)] = cos(u) · du/dx
  • Cosine Rule: d/dx [cos(u)] = -sin(u) · du/dx
  • Exponential Rule: d/dx [e^u] = e^u · du/dx

Calculation Example

Example: Find the derivative of the function f(x) = 4x^5.

Step-by-step solution:

  1. Identify the coefficient: a = 4
  2. Identify the exponent: n = 5
  3. Apply the Power Rule: Multiply the coefficient by the exponent (4 × 5 = 20)
  4. Reduce the exponent by 1: (5 – 1 = 4)
  5. Combine for the final result: f'(x) = 20x^4

Common Questions

What is a derivative?

A derivative represents the sensitivity of a function's output to changes in its input. Geometrically, it is the slope of the tangent line to the graph of the function at a specific point. Our derivative calculator simplifies finding this slope for standard functions.

Can this calculator handle second derivatives?

This specific tool is designed for first-order derivatives. To find a second derivative, you would simply take the result provided by this calculator and input it back into the tool for a second pass.

Why is the derivative of a constant zero?

A constant (like the number 7) does not change regardless of the value of x. Because the derivative measures the rate of change, and a constant has no change, its derivative is always zero.

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