Calculus Examples

$\sin (27 x t) = t$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (27 x t)) = \frac{d}{d x} (t)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 27 x t$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $27 x t$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [27 x t]$

Step 2.1.2

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

$\cos (u) \frac{d}{d x} [27 x t]$

Step 2.1.3

Replace all occurrences of $u$ with $27 x t$ .

$\cos (27 x t) \frac{d}{d x} [27 x t]$

Step 2.2

Since $27$ is constant with respect to $x$ , the derivative of $27 x t$ with respect to $x$ is $27 \frac{d}{d x} [x t]$ .

$\cos (27 x t) (27 \frac{d}{d x} [x t])$

Step 2.3

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 $f (x) = x$ and $g (x) = t$ .

$\cos (27 x t) (27 (x \frac{d}{d x} [t] + t \frac{d}{d x} [x]))$

Step 2.4

Rewrite $\frac{d}{d x} [t]$ as $t'$ .

$\cos (27 x t) (27 (x t' + t \frac{d}{d x} [x]))$

Step 2.5

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

$\cos (27 x t) (27 (x t' + t \cdot 1))$

Step 2.6

Multiply $t$ by $1$ .

$\cos (27 x t) (27 (x t' + t))$

Step 2.7

Simplify.

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Step 2.7.1

Apply the distributive property.

$\cos (27 x t) (27 (x t') + 27 t)$

Step 2.7.2

Apply the distributive property.

$\cos (27 x t) (27 (x t')) + \cos (27 x t) (27 t)$

Step 2.7.3

Reorder terms.

$27 x \cos (27 x t) t' + 27 t \cos (27 x t)$

Step 3

Rewrite $\frac{d}{d x} [t]$ as $t'$ .

$t'$

Step 4

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

$27 x \cos (27 x t) t' + 27 t \cos (27 x t) = t'$

Step 5

Solve for $t'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $27 x \cos (27 x t) t' + 27 t \cos (27 x t)$ .

$27 x t' \cos (27 x t) + 27 t \cos (27 x t) = t'$

Step 5.2

Subtract $t'$ from both sides of the equation.

$27 x t' \cos (27 x t) + 27 t \cos (27 x t) - t' = 0$

Step 5.3

Subtract $27 t \cos (27 x t)$ from both sides of the equation.

$27 x t' \cos (27 x t) - t' = - 27 t \cos (27 x t)$

Step 5.4

Factor $t'$ out of $27 x t' \cos (27 x t) - t'$ .

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Step 5.4.1

Factor $t'$ out of $27 x t' \cos (27 x t)$ .

$t' (27 x \cos (27 x t)) - t' = - 27 t \cos (27 x t)$

Step 5.4.2

Factor $t'$ out of $- t'$ .

$t' (27 x \cos (27 x t)) + t' \cdot - 1 = - 27 t \cos (27 x t)$

Step 5.4.3

Factor $t'$ out of $t' (27 x \cos (27 x t)) + t' \cdot - 1$ .

$t' (27 x \cos (27 x t) - 1) = - 27 t \cos (27 x t)$

Step 5.5

Divide each term in $t' (27 x \cos (27 x t) - 1) = - 27 t \cos (27 x t)$ by $27 x \cos (27 x t) - 1$ and simplify.

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Step 5.5.1

Divide each term in $t' (27 x \cos (27 x t) - 1) = - 27 t \cos (27 x t)$ by $27 x \cos (27 x t) - 1$ .

$\frac{t' (27 x \cos (27 x t) - 1)}{27 x \cos (27 x t) - 1} = \frac{- 27 t \cos (27 x t)}{27 x \cos (27 x t) - 1}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $27 x \cos (27 x t) - 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{t' (27 x \cos (27 x t) - 1)}{27 x \cos (27 x t) - 1} = \frac{- 27 t \cos (27 x t)}{27 x \cos (27 x t) - 1}$

Step 5.5.2.1.2

Divide $t'$ by $1$ .

$t' = \frac{- 27 t \cos (27 x t)}{27 x \cos (27 x t) - 1}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

$t' = - \frac{27 t \cos (27 x t)}{27 x \cos (27 x t) - 1}$

Step 6

Replace $t'$ with $\frac{d t}{d x}$ .

$\frac{d t}{d x} = - \frac{27 t \cos (27 x t)}{27 x \cos (27 x t) - 1}$