Common Base Exponential Differentiation Rules
The first rule is for Common Base Exponential Function, where a is any constant. To obtain the derivative take the natural log of the base (a) and multiply it by the exponent.
DERIVATIVE OF COMMON EXPONENTIAL FUNCTION:
$\frac{d}{dx}\left({a}^{x}\right)=\left(lna\right){a}^{x}$
The second rule is for the natural exponential function, when a = e, where e is the irrational number approximated as 2.718. The derivative of the Natural Exponential Function, e^{x}, is equal to e^{x}.
DERIVATIVE OF NATURAL EXPONENTIAL FUNCTION:
$\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$
Let's take a look at a couple of examples
Step 1: Simplify the expression This expression is already simplified. |
5^{x} + e^{x} |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function as the sum/difference of the derivative of the parts. |
$\frac{d}{dx}\left({5}^{x}+{e}^{x}\right)$ $\frac{d}{dx}{5}^{x}+\frac{d}{dx}{e}^{x}$ |
Step 3: Take the derivative of each part. Use the common exponential rule (CER) to differentiate 5^{x}. Use the natural exponential rule (NER) to differentiate e^{x}. |
$\frac{d}{dx}{5}^{x}=\left(ln5\right){5}^{x}$ CER $\frac{d}{dx}{e}^{x}={e}^{x}$ NER |
Step 4: Add/Subtract the derivatives & simplify. |
$\left(ln5\right){5}^{x}+{e}^{x}$ |
Step 1: Simplify the expression This expression is already simplified. |
6e^{x} + x^{2} - 12^{x} |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function as the sum/difference of the derivative of the parts. |
$\frac{d}{dx}\left(6{e}^{x}+{x}^{2}-{12}^{x}\right)$ $\frac{d}{dx}6{e}^{x}+\frac{d}{dx}{x}^{2}-\frac{d}{dx}{12}^{x}$ |
Step 3: Take the derivative of each part. Use the constant multiple and natural exponential rules (CM/NER) to differentiate 6e^{x}. Use the power rule (PR) to differentiate x^{2}. Use the common exponential rule (CER) to differentiate 12^{x}. |
$\frac{d}{dx}6{e}^{x}=6\frac{d}{dx}{e}^{x}={6}{e}^{x}$ CM/NER $\frac{d}{dx}{x}^{2}=2{x}^{1}={2}{x}$ PR $\frac{d}{dx}{12}^{x}={\left(}{\mathrm{ln}}{12}{\right)}{12}^{x}$ CER |
Step 4: Add/Subtract the derivatives & simplify. |
$6{e}^{x}+2x-\left(\mathrm{ln}12\right){12}^{x}$ |
Step 1: Simplify the expression This expression is already simplified. |
-4e^{x} + 10^{x} |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function as the sum/difference of the derivative of the parts. |
$\frac{d}{dx}\left(-4{e}^{x}+{10}^{x}\right)$ $\frac{d}{dx}-4{e}^{x}+\frac{d}{dx}{10}^{x}$ |
Step 3: Take the derivative of each part. Use the constant multiple and natural exponential rules (CM/NER) to differentiate -4e^{x}. Use the common exponential rule (CER) to differentiate 10^{x}. |
$\frac{d}{dx}-4{e}^{x}=-4\frac{d}{dx}{e}^{x}={-}{4}{e}^{x}$ CM/NER $\frac{d}{dx}{10}^{x}={\left(}{\mathrm{ln}}{10}{\right)}{10}^{x}$ CER |
Step 4: Add/Subtract the derivatives & simplify. |
$-4{e}^{x}+\left(\mathrm{ln}10\right){10}^{x}$ |
Related Links: Math algebra Product Rule Quotient Rule Calculus Topics |
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