Calculus Examples

$f (x) = \tan (x) - x^{3} \cos (x)$

Step 1

By the Sum Rule, the derivative of $\tan (x) - x^{3} \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- x^{3} \cos (x)]$ .

$\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- x^{3} \cos (x)]$

Step 2

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

$\sec^{2} (x) + \frac{d}{d x} [- x^{3} \cos (x)]$

Step 3

Evaluate $\frac{d}{d x} [- x^{3} \cos (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3} \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [x^{3} \cos (x)]$ .

$\sec^{2} (x) - \frac{d}{d x} [x^{3} \cos (x)]$

Step 3.2

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 $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

$\sec^{2} (x) - (x^{3} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x^{3}])$

Step 3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$\sec^{2} (x) - (x^{3} (- \sin (x)) + \cos (x) \frac{d}{d x} [x^{3}])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\sec^{2} (x) - (x^{3} (- \sin (x)) + \cos (x) (3 x^{2}))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\sec^{2} (x) - (x^{3} (- \sin (x))) - (\cos (x) (3 x^{2}))$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$\sec^{2} (x) + 1 (x^{3} (\sin (x))) - (\cos (x) (3 x^{2}))$

Step 4.2.2

Multiply $x^{3}$ by $1$ .

$\sec^{2} (x) + x^{3} \sin (x) - (\cos (x) (3 x^{2}))$

Step 4.2.3

Multiply $3$ by $- 1$ .

$\sec^{2} (x) + x^{3} \sin (x) - 3 \cos (x) x^{2}$

Step 4.3

Reorder terms.

$x^{3} \sin (x) - 3 x^{2} \cos (x) + \sec^{2} (x)$

Step 4.4

Simplify each term.

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Step 4.4.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$x^{3} \sin (x) - 3 x^{2} \cos (x) + {(\frac{1}{\cos (x)})}^{2}$

Step 4.4.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$x^{3} \sin (x) - 3 x^{2} \cos (x) + \frac{1^{2}}{\cos^{2} (x)}$

Step 4.4.3

One to any power is one.

$x^{3} \sin (x) - 3 x^{2} \cos (x) + \frac{1}{\cos^{2} (x)}$

Step 4.5

Simplify each term.

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Step 4.5.1

Rewrite $1$ as $1^{2}$ .

$x^{3} \sin (x) - 3 x^{2} \cos (x) + \frac{1^{2}}{\cos^{2} (x)}$

Step 4.5.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$x^{3} \sin (x) - 3 x^{2} \cos (x) + {(\frac{1}{\cos (x)})}^{2}$

Step 4.5.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$x^{3} \sin (x) - 3 x^{2} \cos (x) + \sec^{2} (x)$