1. Use Power Rule: d/dx(axn) = (a*n)xn-1
2. Multiply "+a+" by "+n+" = "+derCoeff+"
3. Subtract 1 from exponent: "+n+" – 1 = "+derPower+"
4. Result: "+output;}}else if(type==='second'){var derCoeff1=a*n;var derPower1=n-1;var derCoeff2=derCoeff1*derPower1;var derPower2=derPower1-1;var output=derCoeff2+"x"+derPower2+"";if(derPower2===0)output=derCoeff2;if(derPower2===1)output=derCoeff2+"x";resDiv.innerHTML="f"(x) = "+output;if(showSteps){stepDiv.style.display='block';stepDiv.innerHTML="Steps:
1. First Derivative: "+derCoeff1+"x"+derPower1+"
2. Differentiate again: ("+derCoeff1+"*"+derPower1+")x"+derPower1+"-1
3. Result: "+output;}}else if(type==='point'){if(isNaN(c)){alert('Please enter a value for x to evaluate.');return;}var derCoeff=a*n;var derPower=n-1;var slope=derCoeff*Math.pow(c,derPower);resDiv.innerHTML="f'("+c+") = "+slope.toFixed(4);if(showSteps){stepDiv.style.display='block';stepDiv.innerHTML="Steps:
1. Find f'(x): "+derCoeff+"x"+derPower+"
2. Substitute x = "+c+": "+derCoeff+"*("+c+")"+derPower+"
3. Calculate: "+slope;}}if(!showSteps)stepDiv.style.display='none';}
Derivative Calculator Use
This derivative calculator is a specialized tool designed to help students and professionals quickly solve calculus problems involving power functions. By utilizing the Power Rule, the calculator provides instant results for first and second derivatives, as well as the slope of a curve at a specific coordinate.
To use this calculator, simply enter your function's components. Most polynomial terms follow the format axn, where a is the coefficient and n is the exponent. You can select different calculation types from the dropdown menu to match your specific homework or engineering requirement.
- Coefficient (a)
- The constant number that multiplies the variable (e.g., in 5x², the coefficient is 5).
- Exponent (n)
- The power to which the variable is raised (e.g., in 5x², the exponent is 2).
- Evaluate at x (c)
- The specific numeric value where you want to find the instantaneous rate of change or slope of the tangent line.
How It Works
The fundamental principle behind this derivative calculator is the Power Rule of differentiation. The derivative represents the rate at which a function changes at any given point. For any real number exponent, the formula is:
d/dx [axn] = a · n · x(n-1)
- a · n: The new coefficient is the product of the original coefficient and the original exponent.
- x(n-1): The new exponent is the original exponent minus one.
- Second Derivative: This is simply the derivative of the first derivative, representing the curvature or concavity of the function.
- Evaluation: By substituting a value into the derivative expression, we find the numerical slope of the function at that exact point.
Derivative Calculation Example
Example: Find the first derivative of f(x) = 4x3 and evaluate the slope at x = 2.
Step-by-step solution:
- Identify the coefficient: a = 4
- Identify the exponent: n = 3
- Apply the Power Rule: f'(x) = (4 × 3)x(3 – 1)
- Simplify the expression: f'(x) = 12x2
- Substitute x = 2: f'(2) = 12(2)2 = 12(4)
- Result: The slope at x = 2 is 48.
Common Questions
What is a derivative?
In calculus, a derivative measures how a function changes as its input changes. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point. It is a fundamental tool in physics, engineering, and economics to determine rates of change.
Can this derivative calculator handle negative exponents?
Yes, the Power Rule applies to any real number exponent, including negatives and decimals. For example, if you have f(x) = x⁻¹, the derivative is -1x⁻², which is -1/x². Simply enter the negative value into the exponent field.
What happens if the exponent is zero?
If the exponent is zero (f(x) = ax⁰), the function is a constant (f(x) = a). The derivative of any constant is always zero, because a horizontal line has a slope of zero and does not change regardless of the x-value.