Calculus Examples

$y^{6} - 7 x^{5} = 7 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{6} - 7 x^{5}) = \frac{d}{d x} (7 x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $y^{6} - 7 x^{5}$ with respect to $x$ is $\frac{d}{d x} [y^{6}] + \frac{d}{d x} [- 7 x^{5}]$ .

$\frac{d}{d x} [y^{6}] + \frac{d}{d x} [- 7 x^{5}]$

Step 2.2

Evaluate $\frac{d}{d x} [y^{6}]$ .

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = y$ .

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Step 2.2.1.1

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

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [y] + \frac{d}{d x} [- 7 x^{5}]$

Step 2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [y] + \frac{d}{d x} [- 7 x^{5}]$

Step 2.2.1.3

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

$6 y^{5} \frac{d}{d x} [y] + \frac{d}{d x} [- 7 x^{5}]$

Step 2.2.2

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

$6 y^{5} y' + \frac{d}{d x} [- 7 x^{5}]$

Step 2.3

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

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Step 2.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{5}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{5}]$ .

$6 y^{5} y' - 7 \frac{d}{d x} [x^{5}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$6 y^{5} y' - 7 (5 x^{4})$

Step 2.3.3

Multiply $5$ by $- 7$ .

$6 y^{5} y' - 35 x^{4}$

Step 2.4

Reorder terms.

$- 35 x^{4} + 6 y^{5} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 3.2

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

$7 \cdot 1$

Step 3.3

Multiply $7$ by $1$ .

$7$

Step 4

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

$- 35 x^{4} + 6 y^{5} y' = 7$

Step 5

Solve for $y'$ .

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Step 5.1

Add $35 x^{4}$ to both sides of the equation.

$6 y^{5} y' = 7 + 35 x^{4}$

Step 5.2

Divide each term in $6 y^{5} y' = 7 + 35 x^{4}$ by $6 y^{5}$ and simplify.

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Step 5.2.1

Divide each term in $6 y^{5} y' = 7 + 35 x^{4}$ by $6 y^{5}$ .

$\frac{6 y^{5} y'}{6 y^{5}} = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 y^{5} y'}{6 y^{5}} = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y^{5} y'}{y^{5}} = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$

Step 5.2.2.2

Cancel the common factor of $y^{5}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y^{5} y'}{y^{5}} = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$

Step 6

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

$\frac{d y}{d x} = \frac{7}{6 y^{5}} + \frac{35 x^{4}}{6 y^{5}}$