Differentiation of Exponential Functions

Example 1:

$$ y=e^{4x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{4x}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{4x}$$ $$ \textup{Let}\;\; u= 4x$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (4x)$$ $$ \frac{dy}{dx}=e^u(4)$$ $$ \frac{dy}{dx}=4e^u$$ $$ \frac{dy}{dx}=4e^{4x}$$

Example 2:

$$ y=e^{-x},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{-x}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{-x}$$ $$ \textup{Let}\;\; u= -x$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (-x)$$ $$ \frac{dy}{dx}=e^u(-1)$$ $$ \frac{dy}{dx}=-e^u$$ $$ \frac{dy}{dx}=-e^{-x}$$

Example 3:

$$ y=e^{3x+5},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{3x+5}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{3x+5}$$ $$ \textup{Let}\;\; u= 3x+5$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (3x+5)$$ $$ \frac{dy}{dx}=e^u(3)$$ $$ \frac{dy}{dx}=3e^u$$ $$ \frac{dy}{dx}=3e^{3x+5}$$

Example 4:

$$ y=e^{\frac{x}{4}},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{\frac{x}{4}}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{\frac{x}{4}}$$ $$ \textup{Let}\;\; u= \frac{x}{4}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} \left(\frac{x}{4}\right)$$ $$ \frac{dy}{dx}=e^u\left(\frac{1}{4}\right)$$ $$ \frac{dy}{dx}=\frac{1}{4}e^u$$ $$ \frac{dy}{dx}=\frac{1}{4}e^{\frac{x}{4}}$$

Example 5:

$$ y=e^{ax+b},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{ax+b}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{ax+b}$$ $$ \textup{Let}\;\; u= ax+b$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (ax+b)$$ $$ \frac{dy}{dx}=e^u(a)$$ $$ \frac{dy}{dx}=ae^u$$ $$ \frac{dy}{dx}=ae^{ax+b}$$

Example 6:

$$ y=e^{\frac{x}{a}},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{\frac{x}{a}}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{\frac{x}{a}}$$ $$ \textup{Let}\;\; u= \frac{x}{a}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} \left(\frac{x}{a}\right)$$ $$ \frac{dy}{dx}=e^u\left(\frac{1}{a}\right)$$ $$ \frac{dy}{dx}=\frac{1}{a}e^u$$ $$ \frac{dy}{dx}=\frac{1}{a}e^{\frac{x}{a}}$$

Example 7:

$$ y=e^{x^3},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{x^3}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{x^3}$$ $$ \textup{Let}\;\; u= x^3$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (x^3)$$ $$ \frac{dy}{dx}=e^u(3x^2)$$ $$ \frac{dy}{dx}=3x^2e^u$$ $$ \frac{dy}{dx}=3x^2e^{x^3}$$

Example 8:

$$ y=e^{\sqrt{x}},\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=e^{\sqrt{x}}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^{\sqrt{x}}$$ $$ \textup{Let}\;\; u= \sqrt{x}$$ $$ \frac{dy}{dx}=\frac{d}{dx} e^u$$ $$ \frac{dy}{dx}=\frac{d}{du} e^u \frac{du}{dx}$$ $$ \because\;\; \frac{d}{dx} e^x=e^x$$ $$ \therefore\;\; \frac{dy}{dx}=e^u \frac{d}{dx} (\sqrt{x})$$ $$ \frac{dy}{dx}=e^u\left(\frac{1}{2\sqrt{x}}\right)$$ $$ \frac{dy}{dx}=\left(\frac{1}{2\sqrt{x}}\right)e^\sqrt{x}$$ $$ \frac{dy}{dx}=\left(\frac{e^\sqrt{x}}{2\sqrt{x}}\right)$$

Example 9:

$$ y=xe^x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=xe^x$$ $$ \frac{dy}{dx}=\frac{d}{dx} xe^x$$ $$ \because\;\; \frac{d}{dx} uv=u \frac{dv}{dx}+v \frac{du}{dx}$$ $$ \therefore\;\; \frac{dy}{dx}=x\frac{d}{dx}e^x+e^x\frac{d}{dx}x$$ $$ \frac{dy}{dx}=xe^x+e^x(1)$$ $$ \frac{dy}{dx}=xe^x+e^x$$

Example 10:

$$ y=x^4e^x,\;\;\; \textup{Find} \;\;\frac{dy}{dx}$$

Solution:

$$ y=x^4e^x$$ $$ \frac{dy}{dx}=\frac{d}{dx} x^4e^x$$ $$ \because\;\; \frac{d}{dx} uv=u \frac{dv}{dx}+v \frac{du}{dx}$$ $$ \therefore\;\; \frac{dy}{dx}=x^4\frac{d}{dx}e^x+e^x\frac{d}{dx}x^4$$ $$ \frac{dy}{dx}=x^4e^x+e^x(4x^3)$$ $$ \frac{dy}{dx}=x^4e^x+4x^3e^x$$