How do you find the fourth derivative of #e^(-x)#?

Answer 1

#d^4 / (dx^4) f(x) = (-1)^4 * e^(-x) = e^(-x)#

The best way to take a high number of derivatives of a function is to look for a "generating" function. The derivative of an exponential function is the perfect candidate for this since the derivative is just the function itself times the derivative of the exponent (using the chain rule). For example, the first derivative is:

#d / (dx) f(x) = e^(-x) * d/(dx) (-x) = e^(-x)*(-1)#

Taking the derivative again, we get:

#d^2 / (dx^2)f(x) = e^(-x) * (-1) * d/(dx) (-x) = e^(-x)* (-1) * (-1)#
Notice that each time we take the derivative we get a factor of #-1#. We can generalize this to:
#d^n / (dx^n) f(x) = (-1)^n * e^(-x)#
So the 4th derivative (#n=4#) is
#d^4 / (dx^4) f(x) = (-1)^4 * e^(-x) = e^(-x)#
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Answer 2

To find the fourth derivative of ( e^{-x} ), you differentiate the function four times with respect to ( x ). Each time you differentiate, you multiply by the derivative of the exponent, which is always (-1), and then apply the chain rule. After four differentiations, you will find that the fourth derivative is ( e^{-x} ) itself, multiplied by ( (-1)^4 = 1 ). So, the fourth derivative of ( e^{-x} ) is ( e^{-x} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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