# 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 = e ^{x} then dy/dx = e^{x}**

However, a more useful result is this one:

If **y = e ^{f(x)} then dy/dx = f´(x)e^{f(x)}**

**Example 1**

If y = e^{7x-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)e^{f(x)} you will get:

dy/dx = 7e^{7x-4}

**Example 2**

If y = e^{5x+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)e^{f(x)} you will get:

dy/dx = 5e^{5x+2}

**Example 3**

If y = 11e^{4x} 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)e^{f(x)} you will get:

dy/dx = 11 × 4e^{4x} = 44e^{4x}

**Example 4**

If y = e^{cos5x} 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)e^{f(x)} you will get:

dy/dx = -5sin(5x)e^{cos5x}

**Example 5**

If y = e^{sinx} 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)e^{f(x)} you will get:

dy/dx = e^{sinx} 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)e^{f(x)} you will get:

dy/dx = -2 × -9 × e^{-9x} = 18e^{-9x}

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## Comments 1 comment

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)