Calculus Examples

$y = e^{3 x \tan (7 x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{3 x \tan (7 x)})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x \tan (7 x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{1}$ as $3 x \tan (7 x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [3 x \tan (7 x)]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [3 x \tan (7 x)]$

Step 3.1.3

Replace all occurrences of $u_{1}$ with $3 x \tan (7 x)$ .

$e^{3 x \tan (7 x)} \frac{d}{d x} [3 x \tan (7 x)]$

Step 3.2

Differentiate using the Constant Multiple Rule.

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x \tan (7 x)$ with respect to $x$ is $3 \frac{d}{d x} [x \tan (7 x)]$ .

$e^{3 x \tan (7 x)} (3 \frac{d}{d x} [x \tan (7 x)])$

Step 3.2.2

Move $3$ to the left of $e^{3 x \tan (7 x)}$ .

$3 \cdot e^{3 x \tan (7 x)} \frac{d}{d x} [x \tan (7 x)]$

Step 3.3

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 $g (x) = \tan (7 x)$ .

$3 e^{3 x \tan (7 x)} (x \frac{d}{d x} [\tan (7 x)] + \tan (7 x) \frac{d}{d x} [x])$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 7 x$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $7 x$ .

$3 e^{3 x \tan (7 x)} (x (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [7 x]) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.4.2

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

$3 e^{3 x \tan (7 x)} (x (\sec^{2} (u_{2}) \frac{d}{d x} [7 x]) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $7 x$ .

$3 e^{3 x \tan (7 x)} (x (\sec^{2} (7 x) \frac{d}{d x} [7 x]) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.5

Differentiate.

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Step 3.5.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$3 e^{3 x \tan (7 x)} (x (\sec^{2} (7 x) (7 \frac{d}{d x} [x])) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.5.2

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

$3 e^{3 x \tan (7 x)} (x (\sec^{2} (7 x) (7 \cdot 1)) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Multiply $7$ by $1$ .

$3 e^{3 x \tan (7 x)} (x (\sec^{2} (7 x) \cdot 7) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.5.3.2

Move $7$ to the left of $\sec^{2} (7 x)$ .

$3 e^{3 x \tan (7 x)} (x (7 \cdot \sec^{2} (7 x)) + \tan (7 x) \frac{d}{d x} [x])$

Step 3.5.4

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

$3 e^{3 x \tan (7 x)} (x (7 \sec^{2} (7 x)) + \tan (7 x) \cdot 1)$

Step 3.5.5

Multiply $\tan (7 x)$ by $1$ .

$3 e^{3 x \tan (7 x)} (x (7 \sec^{2} (7 x)) + \tan (7 x))$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$3 e^{3 x \tan (7 x)} (x (7 \sec^{2} (7 x))) + 3 e^{3 x \tan (7 x)} \tan (7 x)$

Step 3.6.2

Multiply $7$ by $3$ .

$21 e^{3 x \tan (7 x)} x \sec^{2} (7 x) + 3 e^{3 x \tan (7 x)} \tan (7 x)$

Step 4

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

$y' = 21 e^{3 x \tan (7 x)} x \sec^{2} (7 x) + 3 e^{3 x \tan (7 x)} \tan (7 x)$

Step 5

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

$\frac{d y}{d x} = 21 e^{3 x \tan (7 x)} x \sec^{2} (7 x) + 3 e^{3 x \tan (7 x)} \tan (7 x)$