Algebra Examples

find $\frac{d}{d x} (x^{3} + \sin (x) + 4 \cos (x))$ at $x = π$

Step 1

Replace the variable $x$ with $π$ in the expression.

$\frac{d}{d (π)} \cdot ({(π)}^{3} + \sin (π) + 4 \cos (π))$

Step 2

Cancel the common factor of $d$ .

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Step 2.1

Cancel the common factor.

$\frac{d}{d π} \cdot ({(π)}^{3} + \sin (π) + 4 \cos (π))$

Step 2.2

Rewrite the expression.

$\frac{1}{π} \cdot ({(π)}^{3} + \sin (π) + 4 \cos (π))$

Step 3

Simplify each term.

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Step 3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{π} \cdot (π^{3} + \sin (0) + 4 \cos (π))$

Step 3.2

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

$\frac{1}{π} \cdot (π^{3} + 0 + 4 \cos (π))$

Step 3.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1}{π} \cdot (π^{3} + 0 + 4 (- \cos (0)))$

Step 3.4

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

$\frac{1}{π} \cdot (π^{3} + 0 + 4 (- 1 \cdot 1))$

Step 3.5

Multiply $4 (- 1 \cdot 1)$ .

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Step 3.5.1

Multiply $- 1$ by $1$ .

$\frac{1}{π} \cdot (π^{3} + 0 + 4 \cdot - 1)$

Step 3.5.2

Multiply $4$ by $- 1$ .

$\frac{1}{π} \cdot (π^{3} + 0 - 4)$

Step 4

Simplify terms.

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Step 4.1

Add $π^{3}$ and $0$ .

$\frac{1}{π} \cdot (π^{3} - 4)$

Step 4.2

Apply the distributive property.

$\frac{1}{π} π^{3} + \frac{1}{π} \cdot - 4$

Step 4.3

Cancel the common factor of $π$ .

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Step 4.3.1

Factor $π$ out of $π^{3}$ .

$\frac{1}{π} (π π^{2}) + \frac{1}{π} \cdot - 4$

Step 4.3.2

Cancel the common factor.

$\frac{1}{π} (π π^{2}) + \frac{1}{π} \cdot - 4$

Step 4.3.3

Rewrite the expression.

$π^{2} + \frac{1}{π} \cdot - 4$

Step 4.4

Combine $\frac{1}{π}$ and $- 4$ .

$π^{2} + \frac{- 4}{π}$

Step 4.5

Move the negative in front of the fraction.

$π^{2} - \frac{4}{π}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$π^{2} - \frac{4}{π}$

Decimal Form:

$8.59636485 \dots$