Mastering Differentiation of Standard Functions for JEE

Welcome to this module on differentiating standard functions, a cornerstone of calculus for JEE Mathematics. Understanding these fundamental rules will equip you to tackle a vast array of problems in calculus and its applications.

The Power Rule: The Foundation

The derivative of x^n is nx^(n-1).

The power rule is your first essential tool for differentiation. It tells us how to find the rate of change for any function of the form y = x^n.

The power rule states that if $y = x^n$ , where $n$ is any real number, then its derivative with respect to $x$ , denoted as $\frac{dy}{dx}$ or $f'(x)$ , is given by $\frac{dy}{dx} = nx^{n-1}$ . This rule is fundamental and applies to positive, negative, fractional, and even irrational exponents.

What is the derivative of

x^5

$5x^4$

What is the derivative of

x^{1/2}

(or

\sqrt{x}

$\frac{1}{2}x^{-1/2}$ or $\frac{1}{2\sqrt{x}}$

Derivatives of Trigonometric Functions

Trigonometric functions are ubiquitous in JEE problems. Knowing their derivatives is crucial.

Function	Derivative
$sin(x)$	$cos(x)$
$cos(x)$	$-sin(x)$
$tan(x)$	$sec^2(x)$
$cot(x)$	$-csc^2(x)$
$sec(x)$	$sec(x)tan(x)$
$csc(x)$	$-csc(x)cot(x)$

What is the derivative of

cos(x)

$-sin(x)$

Derivatives of Exponential and Logarithmic Functions

The derivative of e^x is e^x, and the derivative of ln(x) is 1/x.

Exponential and logarithmic functions have remarkably simple derivatives, making them powerful tools in calculus.

The natural exponential function, $y = e^x$ , is unique in that its derivative is itself: $\frac{dy}{dx} = e^x$ . For the natural logarithm function, $y = ln(x)$ , its derivative is $\frac{dy}{dx} = \frac{1}{x}$ for $x > 0$ . These are fundamental for solving problems involving growth, decay, and logarithmic relationships.

What is the derivative of

e^x

$e^x$

What is the derivative of

ln(x)

$\frac{1}{x}$

Constant Multiple and Sum/Difference Rules

These rules allow us to differentiate more complex functions by breaking them down.

Constants can be factored out, and derivatives of sums/differences are sums/differences of derivatives.

Combine basic rules to differentiate combinations of functions. The derivative of a constant times a function is the constant times the derivative of the function. The derivative of a sum or difference of functions is the sum or difference of their derivatives.

If $c$ is a constant and $f(x)$ is a differentiable function, then $\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$ . Furthermore, if $f(x)$ and $g(x)$ are differentiable functions, then $\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$ . These rules are essential for building up to more complex differentiation problems.

What is the derivative of

3x^4

$12x^3$

What is the derivative of

sin(x) + e^x

$cos(x) + e^x$

Putting It All Together: Practice Problems

The key to mastering differentiation of standard functions is consistent practice. Apply the rules learned to various combinations of functions. Remember to identify the structure of the function and apply the appropriate rule.

For JEE, always be prepared for functions that combine these standard forms. Look for patterns and break down complex functions into simpler parts.

Visualizing the derivative of $x^2$ as the slope of the tangent line. As $x$ increases, the slope of the tangent to the parabola $y=x^2$ increases linearly. The power rule $nx^{n-1}$ captures this relationship precisely. For $y=x^2$ , $n=2$ , so the derivative is $2x^{2-1} = 2x$ . This means at $x=1$ , the slope is 2; at $x=2$ , the slope is 4, and so on, reflecting the increasing steepness of the curve.

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Learning Resources

Khan Academy: Derivatives of trigonometric functions(video)

Learn the derivatives of sine, cosine, and tangent with clear explanations and examples.

Brilliant.org: The Power Rule(documentation)

A concise explanation of the power rule, including its derivation and applications.

Paul's Online Math Notes: Derivatives(documentation)

A comprehensive guide to various differentiation rules, including standard functions, product rule, quotient rule, and chain rule.

YouTube: Derivatives of Exponential and Logarithmic Functions(video)

A video tutorial explaining the derivatives of $e^x$ and $ln(x)$ with examples.

Math is Fun: Differentiation Rules(documentation)

An easy-to-understand overview of fundamental differentiation rules, including constant, power, sum, and difference rules.

Coursera: Introduction to Calculus Specialization(tutorial)

A structured course covering the fundamentals of calculus, including differentiation of standard functions.

Stack Exchange Mathematics: Derivative of $x^n$(paper)

Discussions and proofs for the power rule of differentiation, offering deeper insights.

Wikipedia: Derivative(wikipedia)

A broad overview of the concept of derivatives, including their definition and basic rules.

Art of Problem Solving: Calculus(documentation)

A wiki with detailed explanations and problem-solving strategies for calculus topics relevant to competitive exams.

YouTube: Differentiation of Standard Functions (JEE Level)(video)

A practical video focusing on applying differentiation rules to standard functions in the context of JEE preparation. (Note: This is a placeholder for a relevant JEE-focused video, actual link would be specific).