Introduction
Differentiation is a fundamental concept in calculus that deals with the rate at which a function changes at any given point. It is a core topic in the JEE Main Mathematics syllabus and has numerous applications in various fields such as physics, engineering, and economics. This study note will break down the concept of differentiation into digestible parts, ensuring that all nuances are covered comprehensively.
Basic Concepts
Definition of Derivative
The derivative of a function $f(x)$ at a point $x = a$ is defined as the limit:
$$ f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h} $$
This limit, if it exists, gives the slope of the tangent line to the function at $x = a$.
ExampleExample: Find the derivative of the function $f(x) = x^2$ at $x = 3$.
$$ f'(x) = \lim_{{h \to 0}} \frac{(3 + h)^2 - 3^2}{h} = \lim_{{h \to 0}} \frac{9 + 6h + h^2 - 9}{h} = \lim_{{h \to 0}} \frac{6h + h^2}{h} = \lim_{{h \to 0}} (6 + h) = 6 $$
So, $f'(3) = 6$.
Geometric Interpretation
The derivative represents the slope of the tangent line to the curve of the function at a given point. Imagine zooming in on the curve at a point; as you zoom in infinitely, the curve starts looking like a straight line, which is the tangent.
Notation
- $f'(x)$ or $\frac{df}{dx}$ or $\frac{d}{dx}[f(x)]$ are common notations for the derivative of $f(x)$.
- Higher-order derivatives are denoted as $f''(x)$, $f'''(x)$, $\dots$, $f^{(n)}(x)$.
Rules of Differentiation
Power Rule
If $f(x) = x^n$, then:
$$ f'(x) = nx^{n-1} $$
ExampleExample: Differentiate $f(x) = x^5$.
$$ f'(x) = 5x^{4} $$
Sum and Difference Rule
If $f(x) = u(x) + v(x)$ or $f(x) = u(x) - v(x)$, then:
$$ f'(x) = u'(x) + v'(x) \quad \text{or} \quad f'(x) = u'(x) - v'(x) $$
ExampleExample: Differentiate $f(x) = x^3 + 2x^2 - 5x + 7$.
$$ f'(x) = 3x^2 + 4x - 5 $$
Product Rule
If $f(x) = u(x) \cdot v(x)$, then:
$$ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) $$
ExampleExample: Differentiate $f(x) = x^2 \sin(x)$.
$$ f'(x) = 2x \sin(x) + x^2 \cos(x) $$
Quotient Rule
If $f(x) = \frac{u(x)}{v(x)}$, then:
$$ f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{v(x)^2} $$
ExampleExample: Differentiate $f(x) = \frac{x^2}{\cos(x)}$.
$$ f'(x) = \frac{2x \cos(x) + x^2 \sin(x)}{\cos^2(x)} $$
Chain Rule
If $f(x) = g(h(x))$, then:
$$ f'(x) = g'(h(x)) \cdot h'(x) $$
ExampleExample: Differentiate $f(x) = \sin(x^2)$.
Let $u = x^2$. Then $f(x) = \sin(u)$.
$$ f'(x) = \cos(u) \cdot \frac{du}{dx} = \cos(x^2) \cdot 2x = 2x \cos(x^2) $$
Applications of Differentiation
Tangent and Normal Lines
The equation of the tangent line to the curve $y = f(x)$ at the point $(a, f(a))$ is:
$$ y - f(a) = f'(a)(x - a) $$
The equation of the normal line, which is perpendicular to the tangent, is:
$$ y - f(a) = -\frac{1}{f'(a)}(x - a) $$
ExampleExample: Find the equations of the tangent and normal lines to the curve $y = x^2$ at $x = 1$.
- Tangent line: $y - 1 = 2(x - 1)$, which simplifies to $y = 2x - 1$.
- Normal line: $y - 1 = -\frac{1}{2}(x - 1)$, which simplifies to $y = -\frac{1}{2}x + \frac{3}{2}$.
Increasing and Decreasing Functions
A function $f(x)$ is:
- Increasing if $f'(x) > 0$ for all $x$ in an interval.
- Decreasing if $f'(x)
< 0$ for all $x$ in an interval.
NoteTo determine the nature of a function in an interval, analyze the sign of its first derivative.
Maxima and Minima
To find the local maxima and minima:
- Find the critical points by setting $f'(x) = 0$.
- Use the second derivative test:
- If $f''(x) > 0$ at a critical point, it is a local minimum.
- If $f''(x)
< 0$ at a critical point, it is a local maximum.
ExampleExample: Find the local maxima and minima of $f(x) = x^3 - 3x^2 + 2$.
- Find $f'(x) = 3x^2 - 6x$.
- Set $f'(x) = 0$: $3x^2 - 6x = 0 \Rightarrow x(x - 2) = 0 \Rightarrow x = 0, 2$.
- Find $f''(x) = 6x - 6$.
- At $x = 0$: $f''(0) = -6$ (local maximum).
- At $x = 2$: $f''(2) = 6$ (local minimum).
So, there is a local maximum at $x = 0$ and a local minimum at $x = 2$.
TipAlways verify the nature of critical points using the second derivative test or the first derivative test.
Higher-Order Derivatives
The $n$-th derivative of a function $f(x)$ is denoted as $f^{(n)}(x)$ and represents the derivative of the $(n-1)$-th derivative.
ExampleExample: Find the second derivative of $f(x) = x^4 + 2x^3 - x + 7$.
- First derivative: $f'(x) = 4x^3 + 6x^2 - 1$.
- Second derivative: $f''(x) = 12x^2 + 12x$.
A common mistake is to forget to apply the chain rule properly when differentiating composite functions.
Conclusion
Differentiation is a powerful tool in mathematics that helps us understand how functions change. By mastering basic rules and applications, you can solve a wide range of problems in the JEE Main Mathematics syllabus. Practice regularly and pay attention to the nuances of each rule to build a strong foundation in calculus.
