Calculus Examples

$y = - \tan (10 x^{2})$

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan (10 x^{2})$ with respect to $x$ is $- \frac{d}{d x} [\tan (10 x^{2})]$ .

$- \frac{d}{d x} [\tan (10 x^{2})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 10 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $10 x^{2}$ .

$- (\frac{d}{d u} [\tan (u)] \frac{d}{d x} [10 x^{2}])$

Step 2.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$- (\sec^{2} (u) \frac{d}{d x} [10 x^{2}])$

Step 2.3

Replace all occurrences of $u$ with $10 x^{2}$ .

$- (\sec^{2} (10 x^{2}) \frac{d}{d x} [10 x^{2}])$

Step 3

Differentiate.

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [x^{2}]$ .

$- \sec^{2} (10 x^{2}) (10 \frac{d}{d x} [x^{2}])$

Step 3.2

Multiply $10$ by $- 1$ .

$- 10 \sec^{2} (10 x^{2}) \frac{d}{d x} [x^{2}]$

Step 3.3

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

$- 10 \sec^{2} (10 x^{2}) (2 x)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Multiply $2$ by $- 10$ .

$- 20 \sec^{2} (10 x^{2}) x$

Step 3.4.2

Reorder the factors of $- 20 \sec^{2} (10 x^{2}) x$ .

$- 20 x \sec^{2} (10 x^{2})$