Calculus Examples

$7^{x + y} = y^{\ln (7)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7^{x + y}) = \frac{d}{d x} (y^{\ln (7)})$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 7^{x}$ and $g (x) = x + y$ .

Tap for more steps...

Step 2.1.1

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

$\frac{d}{d u} [7^{u}] \frac{d}{d x} [x + y]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $7$ .

$7^{u} \ln (7) \frac{d}{d x} [x + y]$

Step 2.1.3

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

$7^{x + y} \ln (7) \frac{d}{d x} [x + y]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

$7^{x + y} \ln (7) (\frac{d}{d x} [x] + \frac{d}{d x} [y])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$7^{x + y} \ln (7) (1 + \frac{d}{d x} [y])$

Step 2.3

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

$7^{x + y} \ln (7) (1 + y')$

Step 2.4

Reorder the factors of $7^{x + y} \ln (7) (1 + y')$ .

$7^{x + y} (1 + y') \ln (7)$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\ln (7)}$ and $g (x) = y$ .

Tap for more steps...

Step 3.1.1

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

$\frac{d}{d u} [u^{\ln (7)}] \frac{d}{d x} [y]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \ln (7)$ .

$\ln (7) u^{\ln (7) - 1} \frac{d}{d x} [y]$

Step 3.1.3

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

$\ln (7) y^{\ln (7) - 1} \frac{d}{d x} [y]$

Step 3.2

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

$\ln (7) y^{\ln (7) - 1} y'$

Step 3.3

Reorder the factors of $\ln (7) y^{\ln (7) - 1} y'$ .

$y^{\ln (7) - 1} \ln (7) y'$

Step 4

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

$7^{x + y} (1 + y') \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Simplify $7^{x + y} (1 + y') \ln (7)$ .

Tap for more steps...

Step 5.1.1

Rewrite.

$0 + 0 + 7^{x + y} (1 + y') \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5.1.2

Simplify by adding zeros.

$7^{x + y} (1 + y') \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5.1.3

Apply the distributive property.

$(7^{x + y} \cdot 1 + 7^{x + y} y') \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5.1.4

Multiply $7^{x + y}$ by $1$ .

$(7^{x + y} + 7^{x + y} y') \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5.1.5

Apply the distributive property.

$7^{x + y} \ln (7) + 7^{x + y} y' \ln (7) = y^{\ln (7) - 1} \ln (7) y'$

Step 5.1.6

Reorder factors in $7^{x + y} \ln (7) + 7^{x + y} y' \ln (7)$ .

$\ln (7) \cdot 7^{x + y} + y' \ln (7) \cdot 7^{x + y} = y^{\ln (7) - 1} \ln (7) y'$

Step 5.2

Reorder factors in $y^{\ln (7) - 1} \ln (7) y'$ .

$\ln (7) \cdot 7^{x + y} + y' \ln (7) \cdot 7^{x + y} = \ln (7) y' y^{\ln (7) - 1}$

Step 5.3

Subtract $\ln (7) y' y^{\ln (7) - 1}$ from both sides of the equation.

$\ln (7) \cdot 7^{x + y} + y' \ln (7) \cdot 7^{x + y} - \ln (7) y' y^{\ln (7) - 1} = 0$

Step 5.4

Subtract $\ln (7) \cdot 7^{x + y}$ from both sides of the equation.

$y' \ln (7) \cdot 7^{x + y} - \ln (7) y' y^{\ln (7) - 1} = - \ln (7) \cdot 7^{x + y}$

Step 5.5

Factor $y' \ln (7)$ out of $y' \ln (7) \cdot 7^{x + y} - \ln (7) y' y^{\ln (7) - 1}$ .

Tap for more steps...

Step 5.5.1

Factor $y' \ln (7)$ out of $- \ln (7) y' y^{\ln (7) - 1}$ .

$y' \ln (7) \cdot 7^{x + y} + y' \ln (7) (- 1 y^{\ln (7) - 1}) = - \ln (7) \cdot 7^{x + y}$

Step 5.5.2

Factor $y' \ln (7)$ out of $y' \ln (7) \cdot 7^{x + y} + y' \ln (7) (- 1 y^{\ln (7) - 1})$ .

$y' \ln (7) (7^{x + y} - 1 y^{\ln (7) - 1}) = - \ln (7) \cdot 7^{x + y}$

Step 5.6

Rewrite $- 1 y^{\ln (7) - 1}$ as $- y^{\ln (7) - 1}$ .

$y' \ln (7) (7^{x + y} - y^{\ln (7) - 1}) = - \ln (7) \cdot 7^{x + y}$

Step 5.7

Divide each term in $y' \ln (7) (7^{x + y} - y^{\ln (7) - 1}) = - \ln (7) \cdot 7^{x + y}$ by $\ln (7) (7^{x + y} - y^{\ln (7) - 1})$ and simplify.

Tap for more steps...

Step 5.7.1

Divide each term in $y' \ln (7) (7^{x + y} - y^{\ln (7) - 1}) = - \ln (7) \cdot 7^{x + y}$ by $\ln (7) (7^{x + y} - y^{\ln (7) - 1})$ .

$\frac{y' (\ln (7)) (7^{x + y} - y^{\ln (7) - 1})}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})} = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.2

Simplify the left side.

Tap for more steps...

Step 5.7.2.1

Cancel the common factor of $\ln (7)$ .

Tap for more steps...

Step 5.7.2.1.1

Cancel the common factor.

$\frac{y' (\ln (7)) (7^{x + y} - y^{\ln (7) - 1})}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})} = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.2.1.2

Rewrite the expression.

$\frac{y' (7^{x + y} - y^{\ln (7) - 1})}{7^{x + y} - y^{\ln (7) - 1}} = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.2.2

Cancel the common factor of $7^{x + y} - y^{\ln (7) - 1}$ .

Tap for more steps...

Step 5.7.2.2.1

Cancel the common factor.

$\frac{y' (7^{x + y} - y^{\ln (7) - 1})}{7^{x + y} - y^{\ln (7) - 1}} = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.3

Simplify the right side.

Tap for more steps...

Step 5.7.3.1

Cancel the common factor of $\ln (7)$ .

Tap for more steps...

Step 5.7.3.1.1

Cancel the common factor.

$y' = \frac{- \ln (7) \cdot 7^{x + y}}{\ln (7) (7^{x + y} - y^{\ln (7) - 1})}$

Step 5.7.3.1.2

Rewrite the expression.

$y' = \frac{- 1 \cdot 7^{x + y}}{7^{x + y} - y^{\ln (7) - 1}}$

Step 5.7.3.2

Move the negative in front of the fraction.

$y' = - \frac{7^{x + y}}{7^{x + y} - y^{\ln (7) - 1}}$

Step 6

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

$\frac{d y}{d x} = - \frac{7^{x + y}}{7^{x + y} - y^{\ln (7) - 1}}$