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Derivative Calculator & Differentiation Rules Trainer

Calculate 1st, 2nd, and 3rd derivatives of any function with full mathematical steps, formula reasoning, and interactive tangent slope triangle.

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⚙️Function Input (Potenz-, Faktor- & Summenregel)

Degree:

✨Calculated Derivatives

Original Function f(x)
f(x)=3x3−4x2+2x−5f(x) = 3x^{3} - 4x^{2} + 2x - 5
1st Derivative f'(x)
f′(x)=9x2−8x+2f'(x) = 9x^{2} - 8x + 2
Identified Components
Hauptregel:(u+v)′=u′+v′(u + v)' = u' + v'
Potenzregel:(xn)′=n⋅xn−1(x^n)' = n \cdot x^{n-1}

📝Didactic Step-by-Step Solution

Gegebene Funktion: f(x)=3x3−4x2+2x−5\textbf{Gegebene Funktion: } f(x) = 3x^{3} - 4x^{2} + 2x - 5
Angewandte Regeln: Potenzregel (xn)′=n⋅xn−1 und Faktor-/Summenregel\textbf{Angewandte Regeln: } \text{Potenzregel } (x^n)' = n \cdot x^{n-1} \text{ und Faktor-/Summenregel}
Gliedweise Ableitung:\textbf{Gliedweise Ableitung:}
ddx(3x3)=3⋅3x2=9x2\frac{d}{dx}(3x^{3}) = 3 \cdot 3x^{2} = 9x^{2}
ddx(−4x2)=−4⋅2x1=−8x1\frac{d}{dx}(-4x^{2}) = -4 \cdot 2x^{1} = -8x^{1}
ddx(2x1)=2⋅1x0=2x0\frac{d}{dx}(2x^{1}) = 2 \cdot 1x^{0} = 2x^{0}
ddx(−5)=0(Konstantenregel)\frac{d}{dx}(-5) = 0 \quad (\text{Konstantenregel})
Zusammenfassung: f′(x)=9x2−8x+2\textbf{Zusammenfassung: } f'(x) = 9x^{2} - 8x + 2
Tangent Point:
x₀ =1.0
Zoom:
-5-4-3-2-112345-30-25-20-15-10-551015202530P(1.0 | -4.00)m = f'(1.0) = 3.00f'(1.0) = 3.00
Function Graph f(x)
f(1.0) = -4.000
Tangent Equation
y = 3.00·(x - 1.0) - 4.00
Derivative Curve f'(x)
Steigung m = 3.000

Historical Origin & The Challenge Solved

Why was this mathematics developed, where does it come from, and what real-world problem did it solve?

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Epoche & ZeitalterScientific Revolution (17th Century, c. 1665 – 1684)
Pioniere: Sir Isaac Newton, Gottfried Wilhelm Leibniz, Pierre de Fermat

1. The Tangent & Instantaneous Velocity Problem

Before the 17th century, mathematics could only calculate average speeds over time intervals. How could one determine the exact speed of a fired cannonball or a falling apple at a single, infinitesimal moment in time?

2. Differential Calculus & Limits

Newton and Leibniz independently developed differential calculus. By taking the limit as Δt approaches 0 (turning secant slopes into tangent slopes), they created the derivative f'(x) to compute instantaneous rates of change.

3. Artificial Intelligence & Engineering

Without derivatives, deep learning would not exist: training neural networks (ChatGPT, computer vision) relies on gradient descent (backpropagation) to minimize loss across millions of parameters.

„If I have seen further, it is by standing on the shoulders of Giants.“ — Sir Isaac Newton

Praxisnutzen & Reale Anwendungsfälle

Warum lernen wir das? So wird diese Mathematik in Wirtschaft, Technik, Wissenschaft und Alltag praktisch angewendet:

Physics & Kinematics

Velocity & Acceleration

The 1st derivative of position s(t) gives velocity v(t), while the 2nd derivative yields acceleration a(t).

Praxis-Beispiel:

Braking distance analysis in railway engineering to compute stopping trajectories.

Economics & Finance

Marginal Cost & Profit Maximization

Economists utilize derivatives to compute marginal revenue and marginal cost for optimal output pricing.

Praxis-Beispiel:

Finding the optimal production volume under variable economies of scale.

AI & Machine Learning

Gradient Descent & Backpropagation

Neural networks optimize millions of weights using chain-rule derivatives along the loss surface.

Praxis-Beispiel:

Training deep transformers via backward pass gradient propagation.

Mathematical Foundations of Differential Calculus

The derivative f'(x) of a function f represents its instantaneous rate of change and the geometric slope of the tangent line at P(x | f(x)). Formally defined as the limit of the difference quotient: f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h.

Depending on the algebraic structure, standard rules apply: Power rule (x^n)' = n·x^(n-1), Product rule (u·v)' = u'·v + u·v', Quotient rule (u/v)' = (u'·v - u·v') / v², and Chain rule (f(g(x)))' = f'(g(x))·g'(x).

Higher derivatives like f''(x) describe curvature (concavity/convexity) and classify local extrema and inflection points in curve sketching.

Frequently Asked Questions (FAQ)

What is the difference between secant slope and tangent slope?

The secant slope measures the average rate of change between two points, while the tangent slope measures the instantaneous rate of change at a single point.

When should I use the product rule versus the chain rule?

Use the product rule when two separate functions are multiplied, e.g. f(x) = x² · e^x. Use the chain rule when one function is nested inside another, e.g. f(x) = e^(x²).

What does a derivative of f'(x) = 0 indicate?

A horizontal tangent. It is the necessary condition for local extrema (peaks/valleys) and stationary inflection/saddle points.