Calculus Examples

$x = a \cos^{3} (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d ?} (x) = \frac{d}{d ?} (a \cos^{3} (x))$

Step 2

The derivative of $x$ with respect to $?$ is $x'$ .

$x'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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

$a \frac{d}{d ?} [\cos^{3} (x)]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d ?} [f (g (?))]$ is $f' (g (?)) g' (?)$ where $f (?) = ?^{3}$ and $g (?) = \cos (x)$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$a (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d ?} [\cos (x)])$

Step 3.2.2

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

$a (3 {u_{1}}^{2} \frac{d}{d ?} [\cos (x)])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

$a (3 \cos^{2} (x) \frac{d}{d ?} [\cos (x)])$

Step 3.3

Move $3$ to the left of $a$ .

$3 \cdot a \cos^{2} (x) \frac{d}{d ?} [\cos (x)]$

Step 3.4

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

Tap for more steps...

Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$3 a \cos^{2} (x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d ?} [x])$

Step 3.4.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$3 a \cos^{2} (x) (- \sin (u_{2}) \frac{d}{d ?} [x])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $x$ .

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

Step 3.5

Multiply $- 1$ by $3$ .

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

Step 3.6

Rewrite $\frac{d}{d ?} [x]$ as $x'$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$x' = - 3 a \cos^{2} (x) \sin (x) x'$

Step 5

Solve for $x'$ .

Tap for more steps...

Step 5.1

Simplify the right side.

Tap for more steps...

Step 5.1.1

Reorder factors in $- 3 a \cos^{2} (x) \sin (x) x'$ .

$x' = - 3 a x' \cos^{2} (x) \sin (x)$

Step 5.2

Add $3 a x' \cos^{2} (x) \sin (x)$ to both sides of the equation.

$x' + 3 a x' \cos^{2} (x) \sin (x) = 0$

Step 5.3

Factor $x'$ out of $x' + 3 a x' \cos^{2} (x) \sin (x)$ .

Tap for more steps...

Step 5.3.1

Factor $x'$ out of ${x'}^{1}$ .

$x' \cdot 1 + 3 a x' \cos^{2} (x) \sin (x) = 0$

Step 5.3.2

Factor $x'$ out of $3 a x' \cos^{2} (x) \sin (x)$ .

$x' \cdot 1 + x' (3 a \cos^{2} (x) \sin (x)) = 0$

Step 5.3.3

Factor $x'$ out of $x' \cdot 1 + x' (3 a \cos^{2} (x) \sin (x))$ .

$x' (1 + 3 a \cos^{2} (x) \sin (x)) = 0$

Step 5.4

Divide each term in $x' (1 + 3 a \cos^{2} (x) \sin (x)) = 0$ by $1 + 3 a \cos^{2} (x) \sin (x)$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $x' (1 + 3 a \cos^{2} (x) \sin (x)) = 0$ by $1 + 3 a \cos^{2} (x) \sin (x)$ .

$\frac{x' (1 + 3 a \cos^{2} (x) \sin (x))}{1 + 3 a \cos^{2} (x) \sin (x)} = \frac{0}{1 + 3 a \cos^{2} (x) \sin (x)}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $1 + 3 a \cos^{2} (x) \sin (x)$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{x' (1 + 3 a \cos^{2} (x) \sin (x))}{1 + 3 a \cos^{2} (x) \sin (x)} = \frac{0}{1 + 3 a \cos^{2} (x) \sin (x)}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{0}{1 + 3 a \cos^{2} (x) \sin (x)}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Divide $0$ by $1 + 3 a \cos^{2} (x) \sin (x)$ .

$x' = 0$

Step 6

Replace $x'$ with $\frac{d x}{d ?}$ .

$\frac{d x}{d ?} = 0$