# DIFFERENTIATION OF EXPONENTIAL FUNCTIONS AND EULER NUMBER

An exponential function is defined by

$$f(x)=a^x$$

We can differentiate exponential function using the general expression, which is:

$$a^{f(x)}=a^{f(x)}\times \ln(a)\times f'(x)$$

For example, To differentiate $e^x$, it is:

$$\frac{d}{dx}(e^x)=e^x\times \ln(e) \times 1$$

Recalled that $\ln(e)=1$, therefore,

$$\frac{d}{dx}(e^x)=e^x\times 1\times 1=e^x$$

Example 1

Differentiate $10^x$

Solution:

Recalled that the derivative of the exponential function is:

$$a^{f(x)}=a^{f(x)}\times \ln(a)\times f'(x)$$

Here, $a=10$ and $f(x)=x$, Therefore

$$\frac{d}{dx}(10^x)=10^x\times \ln(10)\times 1$$

$$\frac{d}{dx}(10^x)=\ln(10)\times 10^x$$

Example 2

Differentiate $y=e^{4x-1}$

Solution:

Recalled that:

$$a^{f(x)}=a^{f(x)}\times \ln(a)\times f'(x)$$

By the same reasoning,

$$\frac{dy}{dx}(e^{4x-1})=e^{4x-1}\times \ln(e)\times 4$$

If $\ln(e)=1$, then.

$$\frac{dy}{dx}(e^{4x-1})=e^{4x-1}×1×4$$

$$\frac{dy}{dx}(e^{4x-1})=4e^{4x-1}$$

Example 3

If $h(x)=10^{4x^3+3x^2+x}$, Find $h'(x)$

Solution

$$h'(x)=10^{4x^3+3x^2+x}\times \ln(10)\times 12x^2+6x+1$$

$$h'(x)=\ln(10) (12x^2+6x+1)\times 10^{4x^3+3x^2+x}$$

Example 4

Differentiate $y=10^{2x+1}-e^{4x+1}$

Solution:

$$\frac{dy}{dx}=(10^{2x+1}\times \ln(10)×2)-(e^{4x+1}×\ln(e)\times 4)$$

If $\ln(e)=1$

$$\frac{dy}{dx}=2(\ln(10)(10^{2x+1})-(e^{4x+1}\times 1 \times 4)$$

$$\frac{dy}{dx}=2 \ln(10)\times 10^{2x+1}-4e^{4x+1}$$

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Example 5

Differentiate $y=xe^x$.

Solution:

The above is a product. Hence, we use the product rule of differentiation which is exemplified as:

$$(fg)'=f'g+g'f$$

Here, $f=x$, $f'=1$, $g=e^x$, $g'=e^x$

$$\frac{dy}{dx}=1e^x+(e^x)(x)$$

$$\frac{dy}{dx}=e^x(1+x)$$

Voila! We just learned differentiation of exponential. Got questions? tell me in the comment box. Alternatively, You can also ask our telegram community

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