Calculus Examples

$f (x) = x^{10} e^{x} - 7$

Step 1

By the Sum Rule, the derivative of $x^{10} e^{x} - 7$ with respect to $x$ is $\frac{d}{d x} [x^{10} e^{x}] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [x^{10} e^{x}] + \frac{d}{d x} [- 7]$

Step 2

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

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Step 2.1

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 $f (x) = x^{10}$ and $g (x) = e^{x}$ .

$x^{10} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- 7]$

Step 2.2

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

$x^{10} e^{x} + e^{x} \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- 7]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$x^{10} e^{x} + e^{x} (10 x^{9}) + \frac{d}{d x} [- 7]$

Step 3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$x^{10} e^{x} + e^{x} (10 x^{9}) + 0$

Step 4

Simplify.

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Step 4.1

Add $x^{10} e^{x} + e^{x} (10 x^{9})$ and $0$ .

$x^{10} e^{x} + e^{x} (10 x^{9})$

Step 4.2

Reorder terms.

$e^{x} x^{10} + 10 e^{x} x^{9}$

Step 4.3

Reorder factors in $e^{x} x^{10} + 10 e^{x} x^{9}$ .

$x^{10} e^{x} + 10 x^{9} e^{x}$