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Basic Derivative Rules
Understanding the foundational derivative rules is crucial before moving on to more complex functions. These basic rules form the building blocks of calculus differentiation.
Constant Rule
- Rule: The derivative of a constant is zero.
- Mathematical form:
\[ \frac{d}{dx} (c) = 0 \]
- Explanation: Since a constant value does not change with respect to \(x\), its rate of change is zero.
Power Rule
- Rule: The derivative of \(x^n\) where \(n\) is any real number.
- Mathematical form:
\[ \frac{d}{dx} (x^n) = n x^{n - 1} \]
- Application: Used extensively for polynomial functions. For example,
\[ \frac{d}{dx} (x^3) = 3x^2 \]
Constant Multiple Rule
- Rule: The derivative of a constant multiplied by a function.
- Mathematical form:
\[ \frac{d}{dx} [c \cdot f(x)] = c \cdot f'(x) \]
- Application: Simplifies derivations involving constants, e.g.,
\[ \frac{d}{dx} (5x^4) = 5 \cdot 4x^3 = 20x^3 \]
Sum and Difference Rules
- Rules: Derivatives of sums and differences are sums and differences of derivatives.
- Mathematical forms:
\[ \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \]
\[ \frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x) \]
- Application: Useful for differentiating polynomials, sums of functions, etc.
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Product and Quotient Rules
These rules are essential for differentiating functions that are products or quotients of two functions.
Product Rule
- Rule: The derivative of a product of two functions.
- Mathematical form:
\[ \frac{d}{dx} [f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x) \]
- Application: Differentiating functions like \(f(x) = x^2 \sin x\).
Quotient Rule
- Rule: The derivative of a quotient of two functions.
- Mathematical form:
\[ \frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2} \]
- Application: Differentiating functions like \(f(x) = \frac{\ln x}{x^2}\).
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Chain Rule
The chain rule is one of the most powerful tools in differential calculus, used to differentiate composite functions.
Understanding the Chain Rule
- Concept: Differentiating a composition of functions, \(f(g(x))\).
- Mathematical form:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
- Application: Differentiating functions like \(\sqrt{1 + x^2}\), \(e^{3x}\), or \(\sin(2x)\).
Examples of Chain Rule
- For \(f(x) = \sin(3x)\),
\[ f'(x) = \cos(3x) \times 3 = 3 \cos(3x) \]
- For \(f(x) = e^{x^2}\),
\[ f'(x) = e^{x^2} \times 2x = 2x e^{x^2} \]
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Derivative Rules for Special Functions
Certain functions appear frequently in calculus, and knowing their derivatives is essential.
Exponential Functions
- Rule: Derivative of \(e^x\) is itself.
- Mathematical form:
\[ \frac{d}{dx} e^x = e^x \]
- For \(a^x\):
\[ \frac{d}{dx} a^x = a^x \ln a \]
- Application: Derivatives of exponential functions, such as \(e^{2x}\), are computed as
\[ 2 e^{2x} \]
Logarithmic Functions
- Derivative of \(\ln x\):
\[ \frac{d}{dx} \ln x = \frac{1}{x} \]
- Derivative of \(\log_a x\):
\[ \frac{d}{dx} \log_a x = \frac{1}{x \ln a} \]
- Application: Used in derivatives involving logarithmic expressions.
Trigonometric Functions
- \[ \frac{d}{dx} \sin x = \cos x \]
- \[ \frac{d}{dx} \cos x = -\sin x \]
- \[ \frac{d}{dx} \tan x = \sec^2 x \]
- \[ \frac{d}{dx} \cot x = - \csc^2 x \]
- \[ \frac{d}{dx} \sec x = \sec x \tan x \]
- \[ \frac{d}{dx} \csc x = - \csc x \cot x \]
Inverse Trigonometric Functions
- \[ \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \]
- \[ \frac{d}{dx} \arccos x = - \frac{1}{\sqrt{1 - x^2}} \]
- \[ \frac{d}{dx} \arctan x = \frac{1}{1 + x^2} \]
- \[ \frac{d}{dx} \text{arccot} x = - \frac{1}{1 + x^2} \]
- \[ \frac{d}{dx} \text{arcsec} x = \frac{1}{|x| \sqrt{x^2 - 1}} \]
- \[ \frac{d}{dx} \text{arccsc} x = - \frac{1}{|x| \sqrt{x^2 - 1}} \]
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Derivative Rules for Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions with exponential definitions.
- \[ \frac{d}{dx} \sinh x = \cosh x \]
- \[ \frac{d}{dx} \cosh x = \sinh x \]
- \[ \frac{d}{dx} \tanh x = \operatorname{sech}^2 x \]
- \[ \frac{d}{dx} \operatorname{coth} x = - \operatorname{csch}^2 x \]
- \[ \frac{d}{dx} \operatorname{sech} x = - \operatorname{sech} x \tanh x \]
- \[ \frac{d}{dx} \operatorname{csch} x = - \operatorname{csch} x \operatorname{coth} x \]
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Higher-Order Derivatives and Implicit Differentiation
Higher-Order Derivatives
- Definition: Derivatives of derivatives, such as second derivative (\(f''(x)\)), third derivative, etc.
- Notation:
\[ f^{(n)}(x) \]
representing the \(n\)-th derivative.
Implicit Differentiation
- Used when functions are defined implicitly, for example, \(x^2 + y^2 = 25\).
- Technique: Differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\), and solve for \(dy
Frequently Asked Questions
What are the basic derivative rules included in a 'derivative rules cheat sheet'?
The basic derivative rules typically include the power rule, product rule, quotient rule, chain rule, constant rule, and constant multiple rule.
How does the chain rule work in derivative calculations?
The chain rule is used to differentiate composite functions. It states that the derivative of a composition f(g(x)) is f'(g(x)) multiplied by g'(x).
Why is the derivative of a constant zero included in the cheat sheet?
Because the derivative of any constant function is zero, which is fundamental for simplifying derivatives of constant terms.
How can I quickly differentiate products and quotients of functions using the cheat sheet?
Use the product rule (derivative of uv = u'v + uv') and quotient rule (derivative of u/v = (u'v - uv') / v^2) as outlined in the cheat sheet.
What are some common functions with their derivatives that are often included in a cheat sheet?
Common functions include polynomial, exponential, logarithmic, and trigonometric functions, along with their derivatives like d/dx[x^n], e^x, ln(x), sin(x), and cos(x).
How can a cheat sheet help improve my understanding of derivative rules?
A cheat sheet provides quick reference to key rules and formulas, enhancing your ability to solve derivatives efficiently and reinforcing your understanding of fundamental concepts.
