Calculus Examples

$10 x^{5} - 8 x + 6 e^{x} - 10$

Step 1

By the Sum Rule, the derivative of $10 x^{5} - 8 x + 6 e^{x} - 10$ with respect to $x$ is $\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 2

Evaluate $\frac{d}{d x} [10 x^{5}]$ .

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Step 2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{5}$ with respect to $x$ is $10 \frac{d}{d x} [x^{5}]$ .

$10 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$10 (5 x^{4}) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 2.3

Multiply $5$ by $10$ .

$50 x^{4} + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 3

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$50 x^{4} - 8 \frac{d}{d x} [x] + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 3.2

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

$50 x^{4} - 8 \cdot 1 + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 3.3

Multiply $- 8$ by $1$ .

$50 x^{4} - 8 + \frac{d}{d x} [6 e^{x}] + \frac{d}{d x} [- 10]$

Step 4

Evaluate $\frac{d}{d x} [6 e^{x}]$ .

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Step 4.1

Since $6$ is constant with respect to $x$ , the derivative of $6 e^{x}$ with respect to $x$ is $6 \frac{d}{d x} [e^{x}]$ .

$50 x^{4} - 8 + 6 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 10]$

Step 4.2

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

$50 x^{4} - 8 + 6 e^{x} + \frac{d}{d x} [- 10]$

Step 5

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$50 x^{4} - 8 + 6 e^{x} + 0$

Step 6

Simplify.

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Step 6.1

Add $50 x^{4} - 8 + 6 e^{x}$ and $0$ .

$50 x^{4} - 8 + 6 e^{x}$

Step 6.2

Reorder terms.

$50 x^{4} + 6 e^{x} - 8$