Calculus Examples

$f (x) = x^{7} - 7$ , $x = 1.6$

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + \frac{d}{d x} [- 7]$

Step 1.3

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

$7 x^{6} + 0$

Step 1.4

Add $7 x^{6}$ and $0$ .

$7 x^{6}$

Step 2

Replace the variable $x$ with $1.6$ in the expression.

$f' (1.6) = 7 {(1.6)}^{6}$

Step 3

Raise $1.6$ to the power of $6$ .

$f' (1.6) = 7 \cdot 16.777216$

Step 4

Multiply $7$ by $16.777216$ .

$f' (1.6) = 117.440512$