How to Differentiate Calculator

Differentiation Rules:

\[ \frac{d}{dx}(x^n) = nx^{n-1} \] \[ \frac{d}{dx}(e^x) = e^x \] \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \]

Derivative (d/dx f(x)):

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1. What is Differentiation?

Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

2. Basic Differentiation Rules

The most important differentiation rules:

Power Rule: \[ \frac{d}{dx}(x^n) = nx^{n-1} \]

Exponential Rule: \[ \frac{d}{dx}(e^x) = e^x \]

Logarithmic Rule: \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \]

Trigonometric Rules: \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \]

3. How to Use This Calculator

Instructions: Enter a mathematical function in terms of x (or other variable) and select the variable to differentiate with respect to. The calculator will compute the derivative.

4. Common Functions and Their Derivatives

Constant: \( \frac{d}{dx}(c) = 0 \)
Linear: \( \frac{d}{dx}(mx + b) = m \)
Quadratic: \( \frac{d}{dx}(ax^2 + bx + c) = 2ax + b \)
Trigonometric: \( \frac{d}{dx}(\tan x) = \sec^2 x \)
Exponential: \( \frac{d}{dx}(a^x) = a^x \ln a \)

5. Frequently Asked Questions (FAQ)

Q1: What is the chain rule?
A: The chain rule is used for composite functions: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \).

Q2: How do you differentiate implicit functions?
A: Differentiate both sides with respect to x, then solve for dy/dx.

Q3: What is the product rule?
A: \( \frac{d}{dx}(uv) = u'v + uv' \) where u and v are functions of x.

Q4: What is the quotient rule?
A: \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \).

Q5: What are higher order derivatives?
A: These are derivatives of derivatives (second derivative, third derivative, etc.).