Find dy/dx y=1/2x^10-1/3x^3

$y = \frac{1}{2} x^{10} - \frac{1}{3} x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{2} x^{10} - \frac{1}{3} x^{3})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

By the Sum Rule, the derivative of $\frac{1}{2} x^{10} - \frac{1}{3} x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2} x^{10}] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$ .

$\frac{d}{d x} [\frac{1}{2} x^{10}] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{10}]$ .

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{10}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{10}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$\frac{1}{2} (10 x^{9}) + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Combine $10$ and $\frac{1}{2}$ .

$\frac{10}{2} x^{9} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Combine $\frac{10}{2}$ and $x^{9}$ .

$\frac{10 x^{9}}{2} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10 x^{9}$ .

$\frac{2 (5 x^{9})}{2} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 (5 x^{9})}{2 (1)} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Cancel the common factor.

$\frac{2 (5 x^{9})}{2 \cdot 1} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Rewrite the expression.

$\frac{5 x^{9}}{1} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Divide $5 x^{9}$ by $1$ .

$5 x^{9} + \frac{d}{d x} [- \frac{1}{3} x^{3}]$

Evaluate $\frac{d}{d x} [- \frac{1}{3} x^{3}]$ .

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

$5 x^{9} - \frac{1}{3} \frac{d}{d x} [x^{3}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$5 x^{9} - \frac{1}{3} (3 x^{2})$

Multiply $3$ by $- 1$ .

$5 x^{9} - 3 (\frac{1}{3}) x^{2}$

Combine $- 3$ and $\frac{1}{3}$ .

$5 x^{9} + \frac{- 3}{3} x^{2}$

Combine $\frac{- 3}{3}$ and $x^{2}$ .

$5 x^{9} + \frac{- 3 x^{2}}{3}$

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3 x^{2}$ .

$5 x^{9} + \frac{3 (- x^{2})}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$5 x^{9} + \frac{3 (- x^{2})}{3 (1)}$

Cancel the common factor.

$5 x^{9} + \frac{3 (- x^{2})}{3 \cdot 1}$

Rewrite the expression.

$5 x^{9} + \frac{- x^{2}}{1}$

Divide $- x^{2}$ by $1$ .

$5 x^{9} - x^{2}$

Step 4

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

$y' = 5 x^{9} - x^{2}$

Step 5

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

$\frac{d y}{d x} = 5 x^{9} - x^{2}$

Find dy/dx y=1/2x^10-1/3x^3

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