Calculus Examples

$y = 7^{\sec (x)} + \ln (x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (7^{\sec (x)} + \ln (x^{3}))$

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

By the Sum Rule, the derivative of $7^{\sec (x)} + \ln (x^{3})$ with respect to $x$ is $\frac{d}{d x} [7^{\sec (x)}] + \frac{d}{d x} [\ln (x^{3})]$ .

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

Step 3.2

Evaluate $\frac{d}{d x} [7^{\sec (x)}]$ .

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Step 3.2.1

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

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Step 3.2.1.1

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

$\frac{d}{d u_{1}} [7^{u_{1}}] \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\ln (x^{3})]$

Step 3.2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $7$ .

$7^{u_{1}} \ln (7) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\ln (x^{3})]$

Step 3.2.1.3

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

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

Step 3.2.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [\ln (x^{3})]$ .

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Step 3.3.1

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

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Step 3.3.1.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

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

Step 3.3.1.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 3.3.1.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 3.3.2

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

$7^{\sec (x)} \ln (7) \sec (x) \tan (x) + \frac{1}{x^{3}} (3 x^{2})$

Step 3.3.3

Combine $3$ and $\frac{1}{x^{3}}$ .

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

Step 3.3.4

Combine $\frac{3}{x^{3}}$ and $x^{2}$ .

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

Step 3.3.5

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 3.3.5.1

Factor $x^{2}$ out of $3 x^{2}$ .

$7^{\sec (x)} \ln (7) \sec (x) \tan (x) + \frac{x^{2} \cdot 3}{x^{3}}$

Step 3.3.5.2

Cancel the common factors.

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Step 3.3.5.2.1

Factor $x^{2}$ out of $x^{3}$ .

$7^{\sec (x)} \ln (7) \sec (x) \tan (x) + \frac{x^{2} \cdot 3}{x^{2} x}$

Step 3.3.5.2.2

Cancel the common factor.

$7^{\sec (x)} \ln (7) \sec (x) \tan (x) + \frac{x^{2} \cdot 3}{x^{2} x}$

Step 3.3.5.2.3

Rewrite the expression.

$7^{\sec (x)} \ln (7) \sec (x) \tan (x) + \frac{3}{x}$

Step 3.4

Reorder terms.

$7^{\sec (x)} \sec (x) \tan (x) \ln (7) + \frac{3}{x}$

Step 4

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

$y' = 7^{\sec (x)} \sec (x) \tan (x) \ln (7) + \frac{3}{x}$

Step 5

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

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