Introduction
Differentiation is a fundamental concept in calculus that measures how a function changes as its input changes. The derivative of a function provides the rate at which the function is changing at any given point. This concept is widely used in physics, engineering, economics, biology, and many other fields.
Basic Definition
For a function \(f(x)\), the derivative, often denoted by \(f'(x)\) or \(\frac{df}{dx}\), gives the slope of the tangent line to the curve represented by the function at any given point \(x\).
The Difference Quotient:
The derivative is often defined using the difference quotient:
\(f'(x) = \lim_{{h \to 0}} \frac{f(x+h) – f(x)}{h}\)
Basic Differentiation Rules
1. Constant Rule:
The derivative of a constant is zero.
\(\frac{d}{dx}c = 0\) (where \(c\) is a constant)
2. Power Rule:
For any real number \(n\),
\(\frac{d}{dx}x^n = nx^{n-1}\)
3. Constant Multiple Rule:
If \(c\) is a constant and \(f(x)\) is a function, then
\(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\)
4. Sum/Difference Rule:
For functions \(f(x)\) and \(g(x)\),
\(\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\)
5. Product Rule:
For functions \(f(x)\) and \(g(x)\),
\(\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)
6. Quotient Rule:
For functions \(f(x)\) and \(g(x)\) where \(g(x) \neq 0\),
\(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}\)
Some Common Derivatives
- \(\frac{d}{dx} \sin(x) = \cos(x)\)
- \(\frac{d}{dx} \cos(x) = -\sin(x)\)
- \(\frac{d}{dx} \tan(x) = \sec^2(x)\)
- \(\frac{d}{dx} \ln(x) = \frac{1}{x}\)
- \(\frac{d}{dx} e^x = e^x\)
Chain Rule
The chain rule is used to differentiate composite functions. If we have a function \(u(x)\) inside another function \(f(u)\), then:
\(\frac{d}{dx}[f(u(x))] = f'(u) \cdot u'(x)\)
Example:
To differentiate \(f(x) = \sin(3x^2 + 2x)\):
Here, \(u = 3x^2 + 2x\) and \(f(u) = \sin(u)\).
Using the chain rule,
\(f'(x) = \cos(3x^2 + 2x) \cdot (6x + 2)\)
Basic Differentiation of Power Functions:
- Differentiate \(f(x) = x^3 + 4x^2 – 7x\). Using the power rule \(\frac{d}{dx} x^n = nx^{n-1}\):
\(f'(x) = 3x^2 + 8x – 7\)
Differentiate \(g(x) = 2x^4 – 5x^3 + x^2\).
\(g'(x) = 8x^3 – 15x^2 + 2x\)
Differentiate \(h(x) = -x^5 + 3x^3 – 2x\).
\(h'(x) = -5x^4 + 9x^2 – 2\)
Differentiate \(j(x) = x^7 + 4x^6 – x^4 + 3\).
\(j'(x) = 7x^6 + 24x^5 – 4x^3\)
Basic Differentiation for Other Rules:
- Differentiation of Constants: Differentiate \(k(x) = 5\). Since the derivative of a constant is zero:
\(k'(x) = 0\)
Differentiation of Exponential Functions:
Differentiate \(f(x) = e^x\).
Using the rule \(\frac{d}{dx} e^x = e^x\):
\(f'(x) = e^x\)
Differentiation of Trigonometric Functions:
a) Differentiate \(g(x) = \sin(x)\).
Using the rule \(\frac{d}{dx} \sin(x) = \cos(x)\):
\(g'(x) = \cos(x)\)
b) Differentiate \(h(x) = \cos(x)\).
Using the rule \(\frac{d}{dx} \cos(x) = -\sin(x)\):
\(h'(x) = -\sin(x)\)
Differentiation of Natural Logarithm:
Differentiate \(j(x) = \ln(x)\).
Using the rule \(\frac{d}{dx} \ln(x) = \frac{1}{x}\):
\(j'(x) = \frac{1}{x}\)
These examples cover the basic differentiation rules and their application to different types of functions.
Conclusion
Differentiation provides insight into the behavior of functions, allowing us to find maximum and minimum values, analyze rates of change, and solve a myriad of practical problems in various disciplines. As we delve deeper into calculus, we’ll find that differentiation is a powerful tool that underpins many advanced concepts and techniques.