Calculus Examples

$y = 10^{x^{2} - 1}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (10^{x^{2} - 1})$

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 $f (x) = 10^{x}$ and $g (x) = x^{2} - 1$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

$\frac{d}{d u} [10^{u}] \frac{d}{d x} [x^{2} - 1]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $10$ .

$10^{u} \ln (10) \frac{d}{d x} [x^{2} - 1]$

Step 3.1.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

$10^{x^{2} - 1} \ln (10) \frac{d}{d x} [x^{2} - 1]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$10^{x^{2} - 1} \ln (10) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$10^{x^{2} - 1} \ln (10) (2 x + \frac{d}{d x} [- 1])$

Step 3.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$10^{x^{2} - 1} \ln (10) (2 x + 0)$

Step 3.2.4

Add $2 x$ and $0$ .

$10^{x^{2} - 1} \ln (10) (2 x)$

Step 3.3

Simplify.

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Step 3.3.1

Reorder the factors of $10^{x^{2} - 1} \ln (10) (2 x)$ .

$2 \cdot 10^{x^{2} - 1} x \ln (10)$

Step 3.3.2

Reorder factors in $2 \cdot 10^{x^{2} - 1} x \ln (10)$ .

$2 x \cdot 10^{x^{2} - 1} \ln (10)$

Step 4

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

$y' = 2 x \cdot (10^{x^{2} - 1} \ln (10))$

Step 5

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

$\frac{d y}{d x} = 2 x \cdot (10^{x^{2} - 1} \ln (10))$