Calculus Examples

$f (x) = \tan (x) \cos (x)$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \tan (x + h) \cos (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 2.1.2.1.1

Rewrite $\tan (x + h) \cos (x + h)$ in terms of sines and cosines.

$f (x + h) = \frac{\sin (x + h)}{\cos (x + h)} \cdot \cos (x + h)$

Step 2.1.2.1.2

Cancel the common factors.

$f (x + h) = \sin (x + h)$

Step 2.1.2.2

The final answer is $\sin (x + h)$ .

$\sin (x + h)$

Step 2.2

Find the components of the definition.

$f (x + h) = \sin (x + h)$

$f (x) = \sin (x)$

$f (x + h) = \sin (x + h)$

$f (x) = \sin (x)$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\sin (x + h) - (\sin (x))}{h}$

Step 4

Use a sum or difference formula on the numerator.

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Step 4.1

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

$f' (x) = lim_{h \to 0} \frac{(\sin (x) \cos (h) + \cos (x) \sin (h)) - (\sin (x))}{h}$

Step 4.2

Remove parentheses.

$f' (x) = lim_{h \to 0} \frac{(\sin (x) \cos (h) + \cos (x) \sin (h)) - (\sin (x))}{h}$

Step 4.3

Multiply $- 1$ by $\sin (x)$ .

$f' (x) = lim_{h \to 0} \frac{\sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)}{h}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{h \to 0} \sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} \sin (x) \cos (h) + lim_{h \to 0} \cos (x) \sin (h) - lim_{h \to 0} \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.2

Move the term $\sin (x)$ outside of the limit because it is constant with respect to $h$ .

$\frac{\sin (x) lim_{h \to 0} \cos (h) + lim_{h \to 0} \cos (x) \sin (h) - lim_{h \to 0} \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.3

Move the limit inside the trig function because cosine is continuous.

$\frac{\sin (x) \cos (lim_{h \to 0} h) + lim_{h \to 0} \cos (x) \sin (h) - lim_{h \to 0} \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.4

Move the term $\cos (x)$ outside of the limit because it is constant with respect to $h$ .

$\frac{\sin (x) \cos (lim_{h \to 0} h) + \cos (x) lim_{h \to 0} \sin (h) - lim_{h \to 0} \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.5

Move the limit inside the trig function because sine is continuous.

$\frac{\sin (x) \cos (lim_{h \to 0} h) + \cos (x) \sin (lim_{h \to 0} h) - lim_{h \to 0} \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.6

Evaluate the limit of $\sin (x)$ which is constant as $h$ approaches $0$ .

$\frac{\sin (x) \cos (lim_{h \to 0} h) + \cos (x) \sin (lim_{h \to 0} h) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.7

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 5.1.2.7.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sin (x) \cos (0) + \cos (x) \sin (lim_{h \to 0} h) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.7.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sin (x) \cos (0) + \cos (x) \sin (0) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8

Simplify the answer.

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Step 5.1.2.8.1

Simplify each term.

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Step 5.1.2.8.1.1

The exact value of $\cos (0)$ is $1$ .

$\frac{\sin (x) \cdot 1 + \cos (x) \sin (0) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8.1.2

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

$\frac{\sin (x) + \cos (x) \sin (0) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8.1.3

The exact value of $\sin (0)$ is $0$ .

$\frac{\sin (x) + \cos (x) \cdot 0 - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8.1.4

Multiply $\cos (x)$ by $0$ .

$\frac{\sin (x) + 0 - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8.2

Combine the opposite terms in $\sin (x) + 0 - \sin (x)$ .

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Step 5.1.2.8.2.1

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

$\frac{\sin (x) - \sin (x)}{lim_{h \to 0} h}$

Step 5.1.2.8.2.2

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

$\frac{0}{lim_{h \to 0} h}$

Step 5.1.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0}{0}$

Step 5.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 5.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{h \to 0} \frac{\sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [\sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [\sin (x) \cos (h) + \cos (x) \sin (h) - \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.2

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

$lim_{h \to 0} \frac{\frac{d}{d h} [\sin (x) \cos (h)] + \frac{d}{d h} [\cos (x) \sin (h)] + \frac{d}{d h} [- \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.3

Evaluate $\frac{d}{d h} [\sin (x) \cos (h)]$ .

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Step 5.3.3.1

Since $\sin (x)$ is constant with respect to $h$ , the derivative of $\sin (x) \cos (h)$ with respect to $h$ is $\sin (x) \frac{d}{d h} [\cos (h)]$ .

$lim_{h \to 0} \frac{\sin (x) \frac{d}{d h} [\cos (h)] + \frac{d}{d h} [\cos (x) \sin (h)] + \frac{d}{d h} [- \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.3.2

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

$lim_{h \to 0} \frac{\sin (x) (- \sin (h)) + \frac{d}{d h} [\cos (x) \sin (h)] + \frac{d}{d h} [- \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.4

Evaluate $\frac{d}{d h} [\cos (x) \sin (h)]$ .

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Step 5.3.4.1

Since $\cos (x)$ is constant with respect to $h$ , the derivative of $\cos (x) \sin (h)$ with respect to $h$ is $\cos (x) \frac{d}{d h} [\sin (h)]$ .

$lim_{h \to 0} \frac{\sin (x) (- \sin (h)) + \cos (x) \frac{d}{d h} [\sin (h)] + \frac{d}{d h} [- \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.4.2

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

$lim_{h \to 0} \frac{\sin (x) (- \sin (h)) + \cos (x) \cos (h) + \frac{d}{d h} [- \sin (x)]}{\frac{d}{d h} [h]}$

Step 5.3.5

Since $- \sin (x)$ is constant with respect to $h$ , the derivative of $- \sin (x)$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{\sin (x) (- \sin (h)) + \cos (x) \cos (h) + 0}{\frac{d}{d h} [h]}$

Step 5.3.6

Simplify.

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Step 5.3.6.1

Add $\sin (x) (- \sin (h)) + \cos (x) \cos (h)$ and $0$ .

$lim_{h \to 0} \frac{\sin (x) (- \sin (h)) + \cos (x) \cos (h)}{\frac{d}{d h} [h]}$

Step 5.3.6.2

Reorder terms.

$lim_{h \to 0} \frac{- \sin (h) \sin (x) + \cos (h) \cos (x)}{\frac{d}{d h} [h]}$

Step 5.3.7

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

$lim_{h \to 0} \frac{- \sin (h) \sin (x) + \cos (h) \cos (x)}{1}$

Step 5.4

Divide $- \sin (h) \sin (x) + \cos (h) \cos (x)$ by $1$ .

$lim_{h \to 0} - \sin (h) \sin (x) + \cos (h) \cos (x)$

Step 6

Evaluate the limit.

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Step 6.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$- lim_{h \to 0} \sin (h) \sin (x) + lim_{h \to 0} \cos (h) \cos (x)$

Step 6.2

Move the term $\sin (x)$ outside of the limit because it is constant with respect to $h$ .

$- \sin (x) lim_{h \to 0} \sin (h) + lim_{h \to 0} \cos (h) \cos (x)$

Step 6.3

Move the limit inside the trig function because sine is continuous.

$- \sin (x) \sin (lim_{h \to 0} h) + lim_{h \to 0} \cos (h) \cos (x)$

Step 6.4

Move the term $\cos (x)$ outside of the limit because it is constant with respect to $h$ .

$- \sin (x) \sin (lim_{h \to 0} h) + \cos (x) lim_{h \to 0} \cos (h)$

Step 6.5

Move the limit inside the trig function because cosine is continuous.

$- \sin (x) \sin (lim_{h \to 0} h) + \cos (x) \cos (lim_{h \to 0} h)$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 7.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- \sin (x) \sin (0) + \cos (x) \cos (lim_{h \to 0} h)$

Step 7.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$- \sin (x) \sin (0) + \cos (x) \cos (0)$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

The exact value of $\sin (0)$ is $0$ .

$- \sin (x) \cdot 0 + \cos (x) \cos (0)$

Step 8.1.2

Multiply $- \sin (x) \cdot 0$ .

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Step 8.1.2.1

Multiply $0$ by $- 1$ .

$0 \sin (x) + \cos (x) \cos (0)$

Step 8.1.2.2

Multiply $0$ by $\sin (x)$ .

$0 + \cos (x) \cos (0)$

Step 8.1.3

The exact value of $\cos (0)$ is $1$ .

$0 + \cos (x) \cdot 1$

Step 8.1.4

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

$0 + \cos (x)$

Step 8.2

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

$\cos (x)$

Step 9