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Is the derivative of e^x also e^x?
Yes, the derivative of e^x is also e^x. This is because the derivative of e^x with respect to x is simply e^x. In other words, the rate of change of e^x with respect to x is equal to e^x. This property is one of the reasons why the exponential function e^x is so important in calculus and mathematical analysis.
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What is the derivative of e^2?
The derivative of e^2 is 0. This is because the exponential function e^x has a derivative of e^x, and when x is a constant like 2, the derivative is still e^2. Since e is a constant approximately equal to 2.71828, e^2 is also a constant, so its derivative is 0.
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What is the derivative of e^0.5x?
The derivative of e^(0.5x) is 0.5e^(0.5x). This can be found using the chain rule, where the derivative of e^(0.5x) is e^(0.5x) multiplied by the derivative of 0.5x, which is 0.5.
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What is the derivative of e^3?
The derivative of e^3 is 0. This is because the derivative of e^x is simply e^x, and when x is a constant such as 3, the derivative of e^3 is still e^3. Therefore, the derivative of e^3 is 0.
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What is the derivative of e^2x?
The derivative of e^2x is 2e^2x. This can be found using the chain rule, which states that the derivative of e^u is e^u times the derivative of u. In this case, u = 2x, so the derivative of e^2x is e^2x times the derivative of 2x, which is 2. Therefore, the derivative of e^2x is 2e^2x.
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What is the derivative of e-functions?
The derivative of e-functions, such as f(x) = e^x, is simply the e-function itself. In other words, the derivative of e^x is e^x. This is a unique property of e-functions, and it makes them particularly convenient to work with in calculus. When differentiating e-functions, we don't need to apply any special rules or techniques - the derivative is just the original function itself.
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What is the derivative of e^x * 3x?
The derivative of e^x is e^x, and the derivative of 3x is 3. When we have the product of two functions, we can use the product rule to find the derivative. Applying the product rule, the derivative of e^x * 3x is e^x * 3 + 3x * e^x, which simplifies to 3e^x + 3xe^x.
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What is the derivative of e^0.25x^2?
The derivative of e^0.25x^2 is 0.5x*e^0.25x^2. This can be found using the chain rule, where the derivative of the outer function e^0.25x^2 is e^0.25x^2, and the derivative of the inner function 0.25x^2 is 0.5x. Therefore, the derivative of e^0.25x^2 is 0.5x*e^0.25x^2.
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