DIFFERENTIATION OF EXPONENTIAL FUNCTIONS AND EULER NUMBER

Post a Comment

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 $\ln(e)=1$

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

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

Related posts

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

Help us grow our readership by sharing this post

Related Posts

Post a Comment

Subscribe Our Newsletter