Find dx/dy y=x^0.4

$y = x^{0.4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (x^{0.4})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{0.4}$ and $g (y) = x$ .

To apply the Chain Rule, set $u$ as $x$ .

$\frac{d}{d u} [u^{0.4}] \frac{d}{d y} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 0.4$ .

$0.4 u^{- 0.6} \frac{d}{d y} [x]$

Replace all occurrences of $u$ with $x$ .

$0.4 x^{- 0.6} \frac{d}{d y} [x]$

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$0.4 x^{- 0.6} x'$

Simplify.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$0.4 \frac{1}{x^{0.6}} x'$

Combine terms.

Combine $0.4$ and $\frac{1}{x^{0.6}}$ .

$\frac{0.4}{x^{0.6}} x'$

Combine $\frac{0.4}{x^{0.6}}$ and $x'$ .

$\frac{0.4 x'}{x^{0.6}}$

Step 4

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

$1 = \frac{0.4 x'}{x^{0.6}}$

Step 5

Solve for $x'$ .

Rewrite the equation as $\frac{0.4 x'}{x^{0.6}} = 1$ .

$\frac{0.4 x'}{x^{0.6}} = 1$

Multiply both sides by $x^{0.6}$ .

$\frac{0.4 x'}{x^{0.6}} x^{0.6} = 1 x^{0.6}$

Simplify.

Simplify the left side.

Cancel the common factor of $x^{0.6}$ .

Cancel the common factor.

$\frac{0.4 x'}{x^{0.6}} x^{0.6} = 1 x^{0.6}$

Rewrite the expression.

$0.4 x' = 1 x^{0.6}$

Simplify the right side.

Multiply $x^{0.6}$ by $1$ .

$0.4 x' = x^{0.6}$

Divide each term in $0.4 x' = x^{0.6}$ by $0.4$ and simplify.

Divide each term in $0.4 x' = x^{0.6}$ by $0.4$ .

$\frac{0.4 x'}{0.4} = \frac{x^{0.6}}{0.4}$

Simplify the left side.

Cancel the common factor of $0.4$ .

Cancel the common factor.

$\frac{0.4 x'}{0.4} = \frac{x^{0.6}}{0.4}$

Divide $x'$ by $1$ .

$x' = \frac{x^{0.6}}{0.4}$

Simplify the right side.

Multiply by $1$ .

$x' = \frac{1 x^{0.6}}{0.4}$

Factor $0.4$ out of $0.4$ .

$x' = \frac{1 x^{0.6}}{0.4 (1)}$

Separate fractions.

$x' = \frac{1}{0.4} \cdot \frac{x^{0.6}}{1}$

Divide $1$ by $0.4$ .

$x' = 2.5 \frac{x^{0.6}}{1}$

Divide $x^{0.6}$ by $1$ .

$x' = 2.5 x^{0.6}$

Step 6

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

$\frac{d x}{d y} = 2.5 x^{0.6}$

Find dx/dy y=x^0.4

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