Concept question-exponential function derivatives (calc II)

In summary, the conversation discusses the use of implicit differentiation on the left-hand side of the equation y = x^2. After applying ln on both sides, the next step is lny = xlnx. Then, the derivative is found using the product rule, resulting in y'/y = x(1/x) + lnx(1). The explanation for this is that implicit differentiation was used on the lhs. The concept of the chain rule is also mentioned.
  • #1
frasifrasi
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Concept question--exponential function derivatives (calc II)

Ok, so there is an example on my textbook that asks for the derivative of y = x^2.

--after applying ln on both sides, if finally gets to the lny = xlnx step.

But after this step, it just states y'/y = x(1/x)+ lnx(1). I understand it is just the prod. rule on the right, but can anyone explain why it went from lny to y'/y ? Is this a property or formula that they used?

Thank you.
 
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  • #2
they used implicit differentiation on the lhs.

You might remember using it on things like xy=1 or equations like that in Calc I.

so (ln y)' = 1/y * dy/dx which gives the y'/y
 
  • #3
Chain rule?
 
  • #4
Oh, I see. Thank you.
 

FAQ: Concept question-exponential function derivatives (calc II)

1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants. It is characterized by a constantly increasing or decreasing rate of change, depending on the value of b.

2. How do you find the derivative of an exponential function?

To find the derivative of an exponential function, you can use the power rule for derivatives. This means that the derivative of f(x) = ab^x is f'(x) = ab^x ln(b), where ln(b) is the natural logarithm of the base b.

3. What is the significance of the derivative of an exponential function?

The derivative of an exponential function represents the instantaneous rate of change at a specific point on the function. It can also be interpreted as the slope of the tangent line at that point. This is useful in many applications, such as calculating growth rates or predicting future values.

4. How do you use derivatives of exponential functions in real-world problems?

Derivatives of exponential functions can be used in real-world problems to model growth or decay. For example, in finance, they can be used to calculate compound interest or in biology to model population growth. They can also be used in physics and engineering to describe systems with exponentially changing quantities.

5. Can the derivative of an exponential function be negative?

Yes, the derivative of an exponential function can be negative. This occurs when the base b is between 0 and 1, resulting in a decreasing exponential function. In this case, the derivative will also be negative, indicating a decreasing rate of change.

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