Solve ) f(x)= (2x)^ 1 (x^ 2 -36) (x^ 2 - (2x)(x^ 2 -36)^ 2 =- | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$f ( x ) = \frac { ( 2 x ) ^ { \prime } ( x ^ { 2 } - 36 ) - ( 2 x ) ( x ^ { 2 } - 36 ) ^ { \prime } } { ( x ^ { 2 } - 36 ) ^ { 2 } } =$$

Solve for f

$f=-\frac{2\left(x^{2}+36\right)}{x\left(x^{2}-36\right)^{2}}$
$x\neq 0\text{ and }|x|\neq 6$

Steps for Solving Linear Equation

Multiply both sides of the equation by $\left(x-6\right)^{2}\left(x+6\right)^{2}$.

$$fx\left(x-6\right)^{2}\left(x+6\right)^{2}=\frac{\mathrm{d}}{\mathrm{d}x}(2x)\left(x^{2}-36\right)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Use binomial theorem $\left(a-b\right)^{2}=a^{2}-2ab+b^{2}$ to expand $\left(x-6\right)^{2}$.

$$fx\left(x^{2}-12x+36\right)\left(x+6\right)^{2}=\frac{\mathrm{d}}{\mathrm{d}x}(2x)\left(x^{2}-36\right)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(x+6\right)^{2}$.

$$fx\left(x^{2}-12x+36\right)\left(x^{2}+12x+36\right)=\frac{\mathrm{d}}{\mathrm{d}x}(2x)\left(x^{2}-36\right)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Use the distributive property to multiply $fx$ by $x^{2}-12x+36$.

$$\left(fx^{3}-12fx^{2}+36fx\right)\left(x^{2}+12x+36\right)=\frac{\mathrm{d}}{\mathrm{d}x}(2x)\left(x^{2}-36\right)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Use the distributive property to multiply $fx^{3}-12fx^{2}+36fx$ by $x^{2}+12x+36$ and combine like terms.

$$fx^{5}-72fx^{3}+1296fx=\frac{\mathrm{d}}{\mathrm{d}x}(2x)\left(x^{2}-36\right)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Use the distributive property to multiply $\frac{\mathrm{d}}{\mathrm{d}x}(2x)$ by $x^{2}-36$.

$$fx^{5}-72fx^{3}+1296fx=\frac{\mathrm{d}}{\mathrm{d}x}(2x)x^{2}-36\frac{\mathrm{d}}{\mathrm{d}x}(2x)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

Combine all terms containing $f$.

$$\left(x^{5}-72x^{3}+1296x\right)f=\frac{\mathrm{d}}{\mathrm{d}x}(2x)x^{2}-36\frac{\mathrm{d}}{\mathrm{d}x}(2x)-2x\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-36)$$

The equation is in standard form.

$$\left(x^{5}-72x^{3}+1296x\right)f=-2x^{2}-72$$

Divide both sides by $-72x^{3}+1296x+x^{5}$.

$$\frac{\left(x^{5}-72x^{3}+1296x\right)f}{x^{5}-72x^{3}+1296x}=\frac{-2x^{2}-72}{x^{5}-72x^{3}+1296x}$$

Dividing by $-72x^{3}+1296x+x^{5}$ undoes the multiplication by $-72x^{3}+1296x+x^{5}$.

$$f=\frac{-2x^{2}-72}{x^{5}-72x^{3}+1296x}$$

Divide $-2x^{2}-72$ by $-72x^{3}+1296x+x^{5}$.

$$f=-\frac{2\left(x^{2}+36\right)}{x\left(x^{2}-36\right)^{2}}$$