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magma_saber
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Homework Statement
What is the derivative of e^{x3}? also what is the derivative of (ln1/x)^{2}
Homework Equations
The Attempt at a Solution
is it 3x^{2}e^{3x2}?
2(ln1/x)(x)?
Last edited:
Mark44 said:Right for the second one, but I would write it as 2 ln(1/x) (-1/x^2) or even better as
[tex]2 ln(\frac{1}{x}) \frac{-1}{x^2}[/tex]
The general formula for finding the derivative of an exponential function is y = ab^x, where a is the base and b is the constant. The derivative of this function is equal to the natural logarithm of the base, multiplied by the original function. In mathematical notation, this can be written as dy/dx = ln(b) * ab^x.
To use the chain rule to find the derivative of an exponential function, you first need to rewrite the function in the form y = e^u, where u is the exponent. Then, you can apply the chain rule, which states that the derivative of y with respect to x is equal to the derivative of u with respect to x, multiplied by e^u. In other words, dy/dx = du/dx * e^u. Finally, substitute the original exponent back in for u to get the final result.
Yes, the derivative of an exponential function can be negative. This will occur when the base of the exponential function is a fraction less than 1. In this case, the derivative will always be negative, indicating that the function is decreasing as x increases.
The derivative of an exponential function is directly related to the slope of the function's graph at any point. Since the derivative represents the rate of change of the function, a positive derivative indicates a positive slope, meaning the function is increasing. Similarly, a negative derivative indicates a negative slope, meaning the function is decreasing. The derivative also determines the concavity of the function's graph, with a positive derivative indicating a concave up graph and a negative derivative indicating a concave down graph.
No, the derivative of an exponential function is not always an exponential function. In some cases, the derivative may be a polynomial, trigonometric function, or another type of function. The derivative will retain some characteristics of the original exponential function, such as having an exponential term in its equation, but it may also have additional terms depending on the specific function and its variables.