Algebra Examples

$y = {(3 x^{4} - 2 x)}^{- \sin (x)}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${(3 x^{4} - 2 x)}^{- \sin (x)}$ as $e^{\ln ({(3 x^{4} - 2 x)}^{- \sin (x)})}$ .

$\frac{d}{d x} [e^{\ln ({(3 x^{4} - 2 x)}^{- \sin (x)})}]$

Step 1.2

Expand $\ln ({(3 x^{4} - 2 x)}^{- \sin (x)})$ by moving $- \sin (x)$ outside the logarithm.

$\frac{d}{d x} [e^{- \sin (x) \ln (3 x^{4} - 2 x)}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - \sin (x) \ln (3 x^{4} - 2 x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $- \sin (x) \ln (3 x^{4} - 2 x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- \sin (x) \ln (3 x^{4} - 2 x)]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- \sin (x) \ln (3 x^{4} - 2 x)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $- \sin (x) \ln (3 x^{4} - 2 x)$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} \frac{d}{d x} [- \sin (x) \ln (3 x^{4} - 2 x)]$

Step 3

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x) \ln (3 x^{4} - 2 x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x) \ln (3 x^{4} - 2 x)]$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- \frac{d}{d x} [\sin (x) \ln (3 x^{4} - 2 x)])$

Step 4

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 $f (x) = \sin (x)$ and $g (x) = \ln (3 x^{4} - 2 x)$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\sin (x) \frac{d}{d x} [\ln (3 x^{4} - 2 x)] + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x^{4} - 2 x$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $3 x^{4} - 2 x$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\sin (x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [3 x^{4} - 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 5.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\sin (x) (\frac{1}{u_{2}} \frac{d}{d x} [3 x^{4} - 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 5.3

Replace all occurrences of $u_{2}$ with $3 x^{4} - 2 x$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\sin (x) (\frac{1}{3 x^{4} - 2 x} \frac{d}{d x} [3 x^{4} - 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6

Differentiate.

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Step 6.1

Combine $\frac{1}{3 x^{4} - 2 x}$ and $\sin (x)$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} \frac{d}{d x} [3 x^{4} - 2 x] + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.2

By the Sum Rule, the derivative of $3 x^{4} - 2 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x]$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (3 (4 x^{3}) + \frac{d}{d x} [- 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.5

Multiply $4$ by $3$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} + \frac{d}{d x} [- 2 x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.6

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

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2 \frac{d}{d x} [x]) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.7

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

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2 \cdot 1) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 6.8

Multiply $- 2$ by $1$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2) + \ln (3 x^{4} - 2 x) \frac{d}{d x} [\sin (x)]))$

Step 7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2) + \ln (3 x^{4} - 2 x) \cos (x)))$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- (\frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2)) - (\ln (3 x^{4} - 2 x) \cos (x)))$

Step 8.2

Remove parentheses.

$e^{- \sin (x) \ln (3 x^{4} - 2 x)} (- \frac{\sin (x)}{3 x^{4} - 2 x} (12 x^{3} - 2) - \ln (3 x^{4} - 2 x) \cos (x))$