Calculus Examples

$y = e^{7 - (\frac{5}{x})}$

Step 1

Multiply $- 1$ by $\frac{5}{x}$ .

$y = e^{7 - \frac{5}{x}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{7 - \frac{5}{x}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 7 - \frac{5}{x}$ .

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Step 4.1.1

To apply the Chain Rule, set $u$ as $7 - \frac{5}{x}$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [7 - \frac{5}{x}]$

Step 4.1.2

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

$e^{u} \frac{d}{d x} [7 - \frac{5}{x}]$

Step 4.1.3

Replace all occurrences of $u$ with $7 - \frac{5}{x}$ .

$e^{7 - \frac{5}{x}} \frac{d}{d x} [7 - \frac{5}{x}]$

Step 4.2

Differentiate.

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Step 4.2.1

By the Sum Rule, the derivative of $7 - \frac{5}{x}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- \frac{5}{x}]$ .

$e^{7 - \frac{5}{x}} (\frac{d}{d x} [7] + \frac{d}{d x} [- \frac{5}{x}])$

Step 4.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$e^{7 - \frac{5}{x}} (0 + \frac{d}{d x} [- \frac{5}{x}])$

Step 4.2.3

Add $0$ and $\frac{d}{d x} [- \frac{5}{x}]$ .

$e^{7 - \frac{5}{x}} \frac{d}{d x} [- \frac{5}{x}]$

Step 4.2.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x}]$ .

$e^{7 - \frac{5}{x}} (- 5 \frac{d}{d x} [\frac{1}{x}])$

Step 4.2.5

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$e^{7 - \frac{5}{x}} (- 5 \frac{d}{d x} [x^{- 1}])$

Step 4.2.6

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

$e^{7 - \frac{5}{x}} (- 5 (- x^{- 2}))$

Step 4.2.7

Multiply $- 1$ by $- 5$ .

$e^{7 - \frac{5}{x}} (5 x^{- 2})$

Step 4.3

Simplify.

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Step 4.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{7 - \frac{5}{x}} (5 \frac{1}{x^{2}})$

Step 4.3.2

Combine terms.

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Step 4.3.2.1

Combine $5$ and $\frac{1}{x^{2}}$ .

$e^{7 - \frac{5}{x}} \frac{5}{x^{2}}$

Step 4.3.2.2

Combine $e^{7 - \frac{5}{x}}$ and $\frac{5}{x^{2}}$ .

$\frac{e^{7 - \frac{5}{x}} \cdot 5}{x^{2}}$

Step 4.3.2.3

Move $5$ to the left of $e^{7 - \frac{5}{x}}$ .

$\frac{5 e^{7 - \frac{5}{x}}}{x^{2}}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{5 e^{\frac{7 x}{x} - \frac{5}{x}}}{x^{2}}$

Step 4.3.3.2

Combine the numerators over the common denominator.

$\frac{5 e^{\frac{7 x - 5}{x}}}{x^{2}}$

Step 5

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

$y' = \frac{5 e^{\frac{7 x - 5}{x}}}{x^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{5 e^{\frac{7 x - 5}{x}}}{x^{2}}$