Find the derivative of an exponential function - High school Calculus Exercises

Find the derivative of the following functions.

f:x\longmapsto 5^x

f'\left(x\right)=\ln\left(5\right)

f'\left(x\right)=5^{x}

f'\left(x\right)=\dfrac{5^{x}}{\ln\left(5\right)}.

f'\left(x\right)=5^{x}\cdot\ln\left(5\right)

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Here we have:

f'\left(x\right)=\left(5^{x}\right)^{'}=5^{x}\cdot\ln\left(5\right)

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=5^{x}\cdot\ln\left(5\right)

f:x\longmapsto 3\left(\dfrac{1}{7}\right)^{x}

f'\left(x\right)=-\ln\left(7\right)\left(\dfrac{1}{7}\right)^{x}

f'\left(x\right)=-3\ln\left(7\right)\left(\dfrac{1}{7}\right)^{x}

f'\left(x\right)=3\left(\dfrac{1}{7}\right)^{x}

f'\left(x\right)=3x\left(\dfrac{1}{7}\right)^{x-1}.

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Also, if c is a constant and f is a function then:

\left(c\cdot f\right)'=c\cdot\left(f\right)'

Here we have:

f'\left(x\right)=\left(3\left(\dfrac{1}{7}\right)^{x}\right)'=3\cdot\left(\left(\dfrac{1}{7}\right)^{x}\right)'=3\left(\dfrac{1}{7}\right)^{x}\cdot\ln\left(\dfrac{1}{7}\right)=-3\ln\left(7\right)\left(\dfrac{1}{7}\right)^{x}

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=-3\ln\left(7\right)\left(\dfrac{1}{7}\right)^{x}

f:x\longmapsto \left(\ln\left(4\right)\right)^{x}

f'\left(x\right)=\left(\ln\left(4\right)\right)^{x}

f'\left(x\right)=\dfrac{1}{4}\left(\ln\left(4\right)\right)^{x}

f'\left(x\right)=\left(\ln\left(4\right)\right)^{x}\cdot\ln\left(\ln\left(4\right)\right)

f'\left(x\right)=\dfrac{1}{4}\ln\left(4\right)\left(\ln\left(4\right)\right)^{x}

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Here, we have:

f'\left(x\right)=\left(\left(\ln\left(4\right)\right)^{x}\right)'=\left(\ln\left(4\right)\right)^{x}\cdot\ln\left(\ln\left(4\right)\right)

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=\left(\ln\left(4\right)\right)^{x}\cdot\ln\left(\ln\left(4\right)\right).

f:x\longmapsto 5\left(\pi\right)^{2x}

f'\left(x\right)=\ln\left(\pi\right)\cdot\left(\pi\right)^{2x}

f'\left(x\right)=5\cdot\left(\pi\right)^{2x}

f'\left(x\right)=10\cdot\ln\left(\pi\right)\cdot\left(\pi\right)^{2x}

f'\left(x\right)=5\pi\cdot\left(\pi\right)^{2x}

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Also, if c is a constant and f is a function, then:

\left(c\cdot f\right)'=c\cdot\left(f\right)'

If f is a function, then:

\left(a^{f}\right)^{'}=a^{f}\cdot\ln\left(a\right)\cdot\left(f\right)'

Here, we have:

f'\left(x\right)=\left(5\left(\pi\right)^{2x}\right)'=5\cdot\left(\left(\pi\right)^{2x}\right)'=5\cdot\left(\pi\right)^{2x}\ln\left(\pi\right)\cdot2=10\cdot\ln\left(\pi\right)\cdot \left(\pi\right)^{2x}

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=10\cdot\ln\left(\pi\right)\cdot\left(\pi\right)^{2x}.

f:x\longmapsto e^{-x}

f'\left(x\right)=xe^{-x-1}

f'\left(x\right)=e^{x}

f'\left(x\right)=e^{-x}

f'\left(x\right)=- e^{-x}

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Also, if f is a function, then:

\left(a^{f}\right)^{'}=a^{f}\cdot\ln\left(a\right)\cdot\left(f\right)'

Here, we have:

f'\left(x\right)=\left( e^{-x}\right)'= e^{-x}\cdot\left(-x\right)'=- e^{-x}

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=- e^{-x}.

f:x\longmapsto 7e^{3x}

f'\left(x\right)=21\cdot e^{3x-1}

f'\left(x\right)=21\cdot e^{3x}

f'\left(x\right)=7\cdot e^{3x}

f'\left(x\right)=\dfrac{1}{7e^{3x}}

The derivative of an exponential function is:

\left(a^{x}\right)^{'}=a^{x}\cdot\ln\left(a\right)

Also, if c is a constant and f is a function, then:

\left(c\cdot f\right)'=c\cdot\left(f\right)'

If f is a function, then:

\left(a^{f}\right)^{'}=a^{f}\cdot\ln\left(a\right)\cdot\left(f\right)'

Here, we have:

f'\left(x\right)=\left( 7e^{3x}\right)'= 7\cdot\left(e^{3x}\right)'=7\cdot\left(e^{3x}\right)\cdot\ln\left(e\right)\cdot\left(3x\right)'=7\cdot\left(e^{3x}\right)\cdot3=21\cdot e^{3x}

For any x \in \mathbb{R} , f is differentiable and we have:

f'\left(x\right)=21\cdot e^{3x}.