Exponential Differentiation Rules
When functions are added together, the Sum Rule of Differentiation is used. This rule states that the derivative of the sum of the functions is equal to the sum of the derivatives of each part.
SUM RULE OF DIFFERENTIATION:
$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$
When functions are subtracted, the Difference Rule of Differentiation is used. This rule states that the derivative of the difference of the functions is equal to the difference of the derivative of the parts.
DIFFERENCE RULE OF DIFFERENTIATION:
$\frac{d}{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)-\frac{d}{dx}g\left(x\right)$
Let's take a look at a couple of examples
Step 1: Simplify the expression This expression is already simplified. |
x^{3} - 7x + 12 |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function as the sum/difference of the derivative of the parts. |
$\frac{d}{dx}\left({x}^{3}-7x+12\right)$ $\frac{d}{dx}{x}^{3}-\frac{d}{dx}7x+\frac{d}{dx}12$ |
Step 3: Take the derivative of each part. Use the power rule to differentiate x^{3}. Use the constant multiple and power rules to differentiate 7x. Use the constant rule to differentiate 12. |
$\frac{d}{dx}{x}^{3}={3}{{x}}^{{2}}$ Power Rule $\frac{d}{dx}7x=7\frac{d}{dx}x=7{x}^{0}={7}$ CM & Power $\frac{d}{dx}12={0}$ Constant Rule |
Step 4: Add/Subtract the derivatives & simplify. |
3x^{2} - 7 |
Step 1: Simplify the expression Distribute the 2x^{3}. |
6x^{3} - 2x^{4} |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function as the sum/difference of the derivative of the parts. |
$\frac{d}{dx}\left(6{x}^{3}-2{x}^{4}\right)$ $\frac{d}{dx}6{x}^{3}-\frac{d}{dx}2{x}^{4}$ |
Step 3: Take the derivative of each part. Use the constant multiple and power rules to differentiate 6x^{3}. Use the constant multiple and power rules to differentiate 2x^{4}. |
$\frac{d}{dx}6{x}^{3}=6\frac{d}{dx}{x}^{3}={18}{{x}}^{{2}}$ CM & Power $\frac{d}{dx}2{x}^{4}=2\frac{d}{dx}{x}^{4}={8}{{x}}^{{3}}$ CM & Power |
Step 4: Add/Subtract the derivatives & simplify. |
18x^{2} - 8x^{3} |
Step 1: Simplify the expression Rewrite $\frac{1}{3{x}^{2}}$ as $\frac{1}{3}{x}^{-2}$ and remove the parentheses. |
$\frac{d}{dx}\left[\left({x}^{6}+3{x}^{4}-{x}^{3}\right)+\left(\frac{1}{3{x}^{2}}-1\right)\right]$ $\frac{d}{dx}\left[{x}^{6}+3{x}^{4}-{x}^{3}+\frac{1}{3}{x}^{-2}-1\right]$ |
Step 2: Apply the sum/difference rules. Rewrite the derivative of the function and the sum/difference of the derivatives of the parts. |
$\frac{d}{dx}{x}^{6}+\frac{d}{dx}3{x}^{4}-\frac{d}{dx}{x}^{3}+\frac{d}{dx}\frac{1}{3}{x}^{-2}-\frac{d}{dx}1$ |
Step 3: Take the derivative of each part. Use the power rule to differentiate x^{6} Use the constant multiple and power rules to differentiate 3x^{4}. Use the power rule to differentiate x^{3} Use the constant multiple and power rules to differentiate $\frac{1}{3}{x}^{-2}$ Use the constant rule to differentiate 1 |
$\frac{d}{dx}{x}^{6}={6}{{x}}^{{5}}$ Power $\frac{d}{dx}3{x}^{4}=3\frac{d}{dx}{x}^{4}={12}{{x}}^{{3}}$ CM & Power $\frac{d}{dx}{x}^{3}={3}{{x}}^{{2}}$ Power $\frac{d}{dx}\frac{1}{3}{x}^{-2}=\frac{1}{3}\frac{d}{dx}{x}^{-2}={-}\frac{2}{3}{{x}}^{{-}{3}}$ CMPower $\frac{d}{dx}1=0$ Constant |
Step 4: Add/Subtract the derivatives & simplify. |
$6{x}^{5}+12{x}^{3}-3{x}^{2}-\frac{2}{3}{x}^{-3}$ |
Related Links: Math algebra Common Base Exponential Differentiation Rules Product Rule Calculus Topics |
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