Differentiation and Derivatives Questions and Problems with Solutions

Free differentiation questions and problems in calculus are presented along with detailed solutions. Applications of derivatives are also included.
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Differentiation and Derivatives

• Find Derivatives of Functions in Calculus. Find the derivatives of various functions using different methods and rules. Several Examples with detailed solutions are presented. Also exercises with answers are included at the end of the page.
• Derivatives Involving Absolute Value. Examples on how to find the derivative of functions involving absolute value. Exercises with answers are also included.
• Rules of Differentiation of Functions in Calculus. The basic rules of differentiation of functions in calculus are presented along with several examples.
• Find Derivative of y = x^x . A tutorial on how to find the first derivative of y = x^x for x > 0.
• Difference Quotient. We start with the definition of the difference quotient and then use several examples to calculate it. Detailed solutions to questions are presented.
• Use Definition to Find Derivative. The derivative is found using its definition. The difference quotient is first calculated then its limit computed as h ---> 0.
• Proof of Derivative of e^x. The definition of the derivative is used to calculate The derivative of e^x.
• Proof of Derivative of ln(x). The derivative of ln(x) is calculated using the definition.
• Proof of Derivative of sin x. The derivative of sin (x) is calculated using the definition of the derivative as a limit.
• Product Rule of Differentiation with Examples.
• Quotient Rule of Differentiation with Examples.
• Proof of Derivative of a^x.
• Derivative of Logarithm function to Any Base : Log_a (x) .
• Proof of Derivative of cos x. The derivative of cos (x) is calculated using the definition of the derivative as a limit.
• Derivative of tan(x). The derivative of tan (x) is computed using the quotient rule and the derivatives of sin(x) and cos(x).
• Proof of Derivative of cot(x). The proof of the derivative of cot (x) is presented using the quotient rule and the derivatives of sin(x) and cos(x).
• Proof of Derivative of sec(x). The proof of the derivative of sec (x) is presented.
• Proof of Derivative of csc(x). The proof of the derivative of csc (x) is presented.
• Logarithmic Differentiation. A powerful method to find the derivative of complicated functions. The method uses the chain rule and the properties of logarithms.
• Table of Derivatives. A table of derivatives of exponential and logarithmic functions, trigonometric functions and their inverses, hyperbolic functions and their inverses.
• Use the Chain Rule of Differentiation in Calculus. The chain rule of differentiation of functions in calculus is presented along with several examples.
• Implicit Differentiation. Implicit differentiation examples, with detailed solutions, are presented.
• Derivative of Inverse Function. Examples with detailed solutions on how to find the derivative of an inverse function are presented.
• Derivative of Inverse Trigonometric Functions. Formulas of the derivatives of inverse trigonometric functions are presented along with several other examples involving sums, products and quotients of functions.
• Find Derivative of f(x) = arccos(cos(x)) and graph it.
• Find Derivative of f(x) = arcsin(sin(x)) and graph it.
• Find Derivative of f(x) = arctan(tan(x)) and graph it.
• Differentiation of Trigonometric Functions. Formulas of the derivatives of trigonometric functions, in calculus, are presented along with several examples involving products, sums and quotients of trigonometric functions.
• Derivative of a Function Raised to the Power of Another Function.
• Differentiation of Exponential Functions. Formulas and examples of the derivatives of exponential functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.
• Differentiation of Logarithmic Functions. Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.
• Differentiation of Hyperbolic Functions. A table of the derivatives of the hyperbolic functions is presented. Examples, with detailed solutions, involving products, sums, power and quotients of hyperbolic functions are examined.
• Taylor and Maclaurin Series with Examples.

Application of Differentiation

• Absolute Minimum and Maximum of a Function, examples with detailed solutions and graphical interpretations.
• Newton's Method to Find Zeros of a Function. Newton's method is an example of how differentiation is used to find zeros of functions and solve equations numerically. Examples with detailed solutions on how to use Newton's method are presented.
• Linear Approximation of Functions. Linear approximation is another example of how differentiation is used to approximate functions by linear ones close to a given point. Examples with detailed solutions on linear approximations are presented.
• Find Critical Numbers of Functions. Tutorial on how to find the critical numbers of a function. Several examples with detailed solutions and exercises with answers.
• Derivative, Maximum, Minimum of Quadratic Functions. Differentiation is used to analyze the properties such as intervals of increase, decrease, local maximum, local minimum of quadratic functions. Examples with solutions and exercises with answers.
• Determine the Concavity of Quadratic Functions. Examples with solutions and exercises with answers.
• Use Derivative to Show That arcsin(x) + arccos(x) = pi/2.
• Taylor and Maclaurin Series with Examples.

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