Calculus Examples

$y = x \cosh (x) - \sinh (x)$

Step 1

By the Sum Rule, the derivative of $x \cosh (x) - \sinh (x)$ with respect to $x$ is $\frac{d}{d x} [x \cosh (x)] + \frac{d}{d x} [- \sinh (x)]$ .

$\frac{d}{d x} [x \cosh (x)] + \frac{d}{d x} [- \sinh (x)]$

Step 2

Evaluate $\frac{d}{d x} [x \cosh (x)]$ .

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cosh (x)$ .

$x \frac{d}{d x} [\cosh (x)] + \cosh (x) \frac{d}{d x} [x] + \frac{d}{d x} [- \sinh (x)]$

Step 2.2

The derivative of $\cosh (x)$ with respect to $x$ is $\sinh (x)$ .

$x \sinh (x) + \cosh (x) \frac{d}{d x} [x] + \frac{d}{d x} [- \sinh (x)]$

Step 2.3

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

$x \sinh (x) + \cosh (x) \cdot 1 + \frac{d}{d x} [- \sinh (x)]$

Step 2.4

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

$x \sinh (x) + \cosh (x) + \frac{d}{d x} [- \sinh (x)]$

Step 3

Evaluate $\frac{d}{d x} [- \sinh (x)]$ .

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Step 3.1

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

$x \sinh (x) + \cosh (x) - \frac{d}{d x} [\sinh (x)]$

Step 3.2

The derivative of $\sinh (x)$ with respect to $x$ is $\cosh (x)$ .

$x \sinh (x) + \cosh (x) - \cosh (x)$

Step 4

Combine terms.

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Step 4.1

Subtract $\cosh (x)$ from $\cosh (x)$ .

$x \sinh (x) + 0$

Step 4.2

Add $x \sinh (x)$ and $0$ .

$x \sinh (x)$