Calculus Examples

$y = \sinh^{2} (5 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\sinh^{2} (5 x))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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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 $f (x) = x^{2}$ and $g (x) = \sinh (5 x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $\sinh (5 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sinh (5 x)]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\sinh (5 x)]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $\sinh (5 x)$ .

$2 \sinh (5 x) \frac{d}{d x} [\sinh (5 x)]$

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u_{2}$ as $5 x$ .

$2 \sinh (5 x) (\frac{d}{d u_{2}} [\sinh (u_{2})] \frac{d}{d x} [5 x])$

Step 3.2.2

The derivative of $\sinh (u_{2})$ with respect to $u_{2}$ is $\cosh (u_{2})$ .

$2 \sinh (5 x) (\cosh (u_{2}) \frac{d}{d x} [5 x])$

Step 3.2.3

Replace all occurrences of $u_{2}$ with $5 x$ .

$2 \sinh (5 x) (\cosh (5 x) \frac{d}{d x} [5 x])$

Step 3.3

Differentiate.

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Step 3.3.1

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

$2 \sinh (5 x) (\cosh (5 x) (5 \frac{d}{d x} [x]))$

Step 3.3.2

Multiply $5$ by $2$ .

$10 \sinh (5 x) (\cosh (5 x) (\frac{d}{d x} [x]))$

Step 3.3.3

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

$10 \sinh (5 x) (\cosh (5 x) \cdot 1)$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Multiply $\cosh (5 x)$ by $1$ .

$10 \sinh (5 x) \cosh (5 x)$

Step 3.3.4.2

Reorder the factors of $10 \sinh (5 x) \cosh (5 x)$ .

$10 \cosh (5 x) \sinh (5 x)$

Step 4

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

$y' = 10 \cosh (5 x) \sinh (5 x)$

Step 5

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

$\frac{d y}{d x} = 10 \cosh (5 x) \sinh (5 x)$