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

Updated on September 27, 2011

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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