Derivative Calculator with Step-by-Step Solutions
Our derivative calculator solves simple and complex calculus problems instantly. Get accurate dy/dx notation, evaluate the instantaneous rate of change, pinpoint critical points and local extrema, or verify homework using our free online solver.
What is a Derivative in Calculus?
A derivative measures the instantaneous rate of change of a function relative to an independent variable. Geometrically, it calculates the exact slope of the tangent line touching a curve at any coordinate point. Formally, it represents the limit of the average rate of change as the interval approaches zero.
Standard algebra finds average slope over broad intervals. Calculus finds exact slope at a single point. If your first derivative is positive, your curve slopes upward. If negative, it trends downward. When dy/dx equals zero, you have found horizontal tangent slopes where critical points and local extrema live.
The Fundamental Derivative Formulas
The foundational building block of single-variable differentiation is the limit definition of a derivative:
Limit Definition: f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h
Power Rule: d/dx(c * x^n) = c * n * x^(n - 1)
Product Rule: d/dx[u * v] = u'v + uv'
Quotient Rule: d/dx[u / v] = (u'v - uv') / v²
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)When applying our derivative calculator with steps free tool, the engine scans the term structure, isolates coefficients, and combines these fundamental rules to solve multi-term functions.
Step-by-Step Worked Example
Let's evaluate the polynomial: f(x) = 4x³ + 3x² - 5x + 7
Step 1: Apply the Power Rule to Term 1 (4x³). Multiply the exponent 3 by coefficient 4 (giving 12) and subtract 1 from the exponent. Result: 12x².
Step 2: Apply the Power Rule to Term 2 (3x²). Multiply the exponent 2 by 3 (giving 6) and subtract 1 from the power. Result: 6x.
Step 3: Differentiate Linear Term 3 (-5x). For x raised to the power of 1, the derivative leaves only the standalone coefficient. Result: -5.
Step 4: Differentiate Constant Term 4 (+7). A constant value never changes, meaning its instantaneous rate of change is zero. Result: 0.
Step 5: Write the First Derivative (dy/dx). Combine each term: f'(x) = 12x² + 6x - 5.
Advanced Calculus Operations
Real-world engineering, financial modeling, and machine learning architectures require analytical operations beyond single polynomial differentiation:
- Second Derivative Calculator Step by Step:Differentiating f'(x) yields the second derivative, f''(x) = 24x + 6. This describes concavity, tracks acceleration over velocity, and detects graph inflection points.
- Implicit Differentiation Calculator Methods: When equations bind x and y together (like x² + y² = 25), implicit differentiation applies the chain rule with respect to x across all terms containing y, solving directly for dy/dx.
- Partial Derivative Calculator Online Analysis: In multivariable calculus, partial derivatives (∂f/∂x and ∂f/∂y) measure how a surface slopes in one specific directional axis while holding all other variables constant.
Frequently Asked Questions
How do you find the derivative of a function?
Identify the operational setup of your terms. Apply the power rule for exponents, product and quotient rules for multiplied or divided expressions, and the chain rule for composite functions. You can also evaluate the formal limit definition of a derivative as interval h approaches zero.
What is the derivative of ln(x)?
The derivative of the natural log function f(x) = ln(x) is 1/x for all x > 0. When evaluating composite logarithmic terms like ln(g(x)), use the chain rule to obtain g'(x) / g(x).
What is the difference between a derivative and a partial derivative?
A standard derivative (dy/dx) measures the rate of change in a single-variable system. A partial derivative calculates the slope of a multivariable function along one chosen axis while treating every other variable as a fixed constant.
How do you calculate the second derivative?
Differentiate the original function to get your first derivative f'(x), then differentiate that resulting expression a second time to produce f''(x) (d²y/dx²). This reveals the concavity of the curve and identifies points of inflection.
Is the derivative calculator accurate for trig functions?
Yes. Analytical solvers evaluate basic trigonometric functions (like sin, cos, and tan) as well as inverse trigonometric and composite expressions using the chain rule with exact mathematical accuracy.
Derivative Calculator with Steps
• d/dx [3x^3] = 3 × 3x^(3-1) = 9x^2
• d/dx [4x^2] = 4 × 2x^(2-1) = 8x
• d/dx [5x] = 5 × 1x^(1-1) = 5
• d/dx [6] = 0 (Constant Rule)
* Derivative computations follow formal calculus differentiation power rules. First derivatives measure the instantaneous velocity/gradient, while second derivatives evaluate the geometric concavity and inflection behavior of the curve.
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