How to Differentiate Negative Exponentials

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Differentiation is one of the key components of calculus. Differentiation is a mathematical process for discovering how a mathematical function changes at a particular instant in time. This process can be applied to many different types of functions, including the exponential function (y = e^x, in mathematical terms), which has a particularly important place in calculus, as the function remains the same when differentiated. Negative exponentials (that is, an exponential taken to a negative power) are a special case of this process, but are relatively straightforward to calculate.

    Write down the function you will be differentiating. As an example, assume the function is e to the negative x, or y = e^(-x).

    Differentiate the equation. This question is an example of the chain rule in calculus, where one function is located within another function; in mathematical notation, this is written as f(g(x)), where g(x) is a function within the function f. The chain rule is written as

    y' = f'(g(x)) * g'(x),

    where the ' indicates differentiation and * indicates multiplication. Therefore, differentiate the function in the exponent and multiply this by the original exponent. In equation form, this is written as y = e^[f(x)]*f'(x)

    Applying this to the function y = e(-x) gives the equation y' = e^x *(-1), since the derivative of -x is -1 and the derivative of e^x is e^x.

    Simplify the differentiated function:

    y = e^(-x) * (-1) gives y = -e^(-x).

    Therefore, this is the derivative of the negative exponential.


About the Author

Thomas Bourdin began writing professionally in 2010. He writes for various websites, where his interests include science, computers and music. He holds a Bachelor of Science degree in physics with a minor in mathematics from the University of Saskatchewan and a Master of Science in physics from Ryerson University.