Calculus Examples

$y = 9^{1 - x^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (9^{1 - x^{2}})$

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 $f (x) = 9^{x}$ and $g (x) = 1 - x^{2}$ .

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Step 3.1.1

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

$\frac{d}{d u} [9^{u}] \frac{d}{d x} [1 - x^{2}]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $9$ .

$9^{u} \ln (9) \frac{d}{d x} [1 - x^{2}]$

Step 3.1.3

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

$9^{1 - x^{2}} \ln (9) \frac{d}{d x} [1 - x^{2}]$

Step 3.2

Differentiate.

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Step 3.2.1

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

$9^{1 - x^{2}} \ln (9) (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}])$

Step 3.2.2

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

$9^{1 - x^{2}} \ln (9) (0 + \frac{d}{d x} [- x^{2}])$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$9^{1 - x^{2}} \ln (9) \frac{d}{d x} [- x^{2}]$

Step 3.2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$9^{1 - x^{2}} \ln (9) (- \frac{d}{d x} [x^{2}])$

Step 3.2.5

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

$9^{1 - x^{2}} \ln (9) (- (2 x))$

Step 3.2.6

Multiply $2$ by $- 1$ .

$9^{1 - x^{2}} \ln (9) (- 2 x)$

Step 3.3

Simplify.

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Step 3.3.1

Reorder the factors of $9^{1 - x^{2}} \ln (9) (- 2 x)$ .

$- 2 \cdot 9^{1 - x^{2}} x \ln (9)$

Step 3.3.2

Reorder factors in $- 2 \cdot 9^{1 - x^{2}} x \ln (9)$ .

$- 2 x \cdot 9^{1 - x^{2}} \ln (9)$

Step 4

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

$y' = - 2 x \cdot (9^{1 - x^{2}} \ln (9))$

Step 5

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

$\frac{d y}{d x} = - 2 x \cdot (9^{1 - x^{2}} \ln (9))$