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Questions and Answers
What is the derivative of ln x?
What is the derivative of ln x?
1/x
What is the differentiation formula for y = ln u?
What is the differentiation formula for y = ln u?
1/u * du/dx
How do you differentiate the function y = ln(2x + 5)?
How do you differentiate the function y = ln(2x + 5)?
3/(2x + 5)
What does the change-of-base formula for logarithms state?
What does the change-of-base formula for logarithms state?
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How do you differentiate the function y = log2 x?
How do you differentiate the function y = log2 x?
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What is the derivative of an exponential function e^u?
What is the derivative of an exponential function e^u?
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What is the inverse function rule for differentiating f(f^-1(x))?
What is the inverse function rule for differentiating f(f^-1(x))?
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Study Notes
Derivatives Of Logarithmic Functions
- The derivative of
ln xis1/xforx > 0 - The derivative of
ln uis(1/u) * (du/dx)foru ≠ 0where u is a differentiable function of x
Derivatives Of Exponential Functions
- The derivative of
e^uise^u * (du/dx)where u is a differential function of x - This formula is derived from the inverse function rule, which states that the derivative of the inverse of a function is the reciprocal of the derivative of the original function
Elasticity of Demand
- Elasticity of demand is an economic measure of the sensitivity of the quantity demanded of a good to changes in its price
- The elasticity of demand is the ratio of the percentage change in quantity demanded to the percentage change in price
Implicit Differentiation
- Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly, i.e. a function where the dependent variable is not explicitly expressed in terms of the independent variable.
- The method involves differentiating both sides of the equation with respect to the independent variable, treating the dependent variable as a function of the independent variable, and then solving for the derivative
Logarithmic Differentiation
- Logarithmic differentiation is a technique used to find the derivative of a function that is expressed as a product, quotient, or power of other functions.
- The method involves taking the natural logarithm of both sides of the equation, differentiating both sides, and then solving for the derivative
- This technique makes it easier to differentiate complicated expressions using logarithm properties
Higher-Order Derivatives
- Higher-order derivatives are derivatives of derivatives
- The second derivative is the derivative of the first derivative
- The third derivative is the derivative of the second derivative, and so on
- These derivatives are useful in many areas of mathematics including calculus, differential equations and physics
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Description
This quiz covers essential concepts in calculus related to derivatives of logarithmic and exponential functions, along with the elasticity of demand in economics. Understand implicit differentiation techniques and how they apply to various functions. Test your knowledge and comprehension of these fundamental principles.
