Differentiating the exponential function. The derivative of e^x.


The exponential function is probably one of the easiest functions to differentiate as the derivative is the same function. This means that the derivative of exponential is exponential:

So if y = ex then dy/dx = ex

However, a more useful result is this one:

If y = ef(x) then dy/dx = f´(x)ef(x)

Example 1

If y = e7x-4 then work out the derivative of y.

So in this question f(x) = 7x -4 so f´(x) = 7. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = 7e7x-4

Example 2

If y = e5x+2 then work out the derivative of y.

So in this question f(x) = 5x + 2 so f´(x) = 5. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = 5e5x+2

Example 3

If y = 11e4x then work out the derivative of y.

So in this question f(x) = 4x so f´(x) = 4. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = 11 × 4e4x = 44e4x

Example 4

If y = ecos5x then work out the derivative of y.

So in this question f(x) = cos5x so f´(x) = -5sin5x. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = -5sin(5x)ecos5x

Example 5

If y = esinx then work out the derivative of y.

So in this question f(x) = sinx so f´(x) = cosx. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = esinx cosx

Example 6

If y = -2e-9x then work out the derivative of y.

So in this question f(x) = -9x so f´(x) = -9. Also you now that the derivative of exponential is exponential.

So using dy/dx = f´(x)ef(x) you will get:

dy/dx = -2 × -9 × e-9x = 18e-9x

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academysigma 4 years ago from Hong Kong

Thanks for this. I'd like to add that this is the chain rule in action and can be applied anytime you have a function of a function (i.e. composite functions) so if:

h(x) = f(g(x)), then h'(x) = f'(g(x)).g'(x)

Simple polynomial example:

h(x) = (3x^2 + 10)^5, then h'(x) = 5(3x^2 + 10)^4 . (6x)

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