Calculus Examples

$5^{x} + 5^{y} = 10$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5^{x} + 5^{y}) = \frac{d}{d x} (10)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $5^{x} + 5^{y}$ with respect to $x$ is $\frac{d}{d x} [5^{x}] + \frac{d}{d x} [5^{y}]$ .

$\frac{d}{d x} [5^{x}] + \frac{d}{d x} [5^{y}]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$5^{x} \ln (5) + \frac{d}{d x} [5^{y}]$

Step 2.3

Evaluate $\frac{d}{d x} [5^{y}]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 5^{x}$ and $g (x) = y$ .

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Step 2.3.1.1

To apply the Chain Rule, set $u$ as $y$ .

$5^{x} \ln (5) + \frac{d}{d u} [5^{u}] \frac{d}{d x} [y]$

Step 2.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $5$ .

$5^{x} \ln (5) + 5^{u} \ln (5) \frac{d}{d x} [y]$

Step 2.3.1.3

Replace all occurrences of $u$ with $y$ .

$5^{x} \ln (5) + 5^{y} \ln (5) \frac{d}{d x} [y]$

Step 2.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$5^{x} \ln (5) + 5^{y} \ln (5) y'$

Step 3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$5^{x} \ln (5) + 5^{y} \ln (5) y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $5^{x} \ln (5) + 5^{y} \ln (5) y'$ .

$5^{x} \ln (5) + y' \cdot (5^{y} \ln (5)) = 0$

Step 5.1.2

Reorder $y'$ and $5^{y}$ .

$5^{x} \ln (5) + 5^{y} \cdot (y' \ln (5)) = 0$

Step 5.1.3

Reorder $5^{x} \ln (5)$ and $5^{y} \cdot y' \ln (5)$ .

$5^{y} \cdot (y' \ln (5)) + 5^{x} \ln (5) = 0$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify $5^{y} \cdot (y' \ln (5)) + 5^{x} \ln (5)$ .

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Step 5.2.1.1

Remove parentheses.

$5^{y} y' \ln (5) + 5^{x} \ln (5) = 0$

Step 5.2.1.2

Reorder factors in $5^{y} y' \ln (5) + 5^{x} \ln (5)$ .

$y' \ln (5) \cdot 5^{y} + \ln (5) \cdot 5^{x} = 0$

Step 5.3

Move all the terms containing a logarithm to the left side of the equation.

$y' \ln (5) + \ln (5) = - 5^{y} - 5^{x} + 0$

Step 5.4

Add $- 5^{y} - 5^{x}$ and $0$ .

$y' \ln (5) + \ln (5) = - 5^{y} - 5^{x}$

Step 5.5

Subtract $\ln (5)$ from both sides of the equation.

$y' \ln (5) = - 5^{y} - 5^{x} - \ln (5)$

Step 5.6

Divide each term in $y' \ln (5) = - 5^{y} - 5^{x} - \ln (5)$ by $\ln (5)$ and simplify.

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Step 5.6.1

Divide each term in $y' \ln (5) = - 5^{y} - 5^{x} - \ln (5)$ by $\ln (5)$ .

$\frac{y' \ln (5)}{\ln (5)} = \frac{- 5^{y}}{\ln (5)} + \frac{- 5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $\ln (5)$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{y' \ln (5)}{\ln (5)} = \frac{- 5^{y}}{\ln (5)} + \frac{- 5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5^{y}}{\ln (5)} + \frac{- 5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{5^{y}}{\ln (5)} + \frac{- 5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{5^{y}}{\ln (5)} - \frac{5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.3.1.3

Cancel the common factor of $\ln (5)$ .

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Step 5.6.3.1.3.1

Cancel the common factor.

$y' = - \frac{5^{y}}{\ln (5)} - \frac{5^{x}}{\ln (5)} + \frac{- \ln (5)}{\ln (5)}$

Step 5.6.3.1.3.2

Divide $- 1$ by $1$ .

$y' = - \frac{5^{y}}{\ln (5)} - \frac{5^{x}}{\ln (5)} - 1$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{5^{y}}{\ln (5)} - \frac{5^{x}}{\ln (5)} - 1$