Solve x/x+8/x^3

$$\frac { \frac { x } { x + 8 } } { x ^ { 3 } }$$

Evaluate

$$\frac{1}{\left(x+8\right)x^{2}}$$

Solution Steps

Express \frac{\frac{x}{x+8}}{x^{3}} as a single fraction.

$$\frac{x}{\left(x+8\right)x^{3}}$$

Cancel out x in both numerator and denominator.

$$\frac{1}{\left(x+8\right)x^{2}}$$

Use the distributive property to multiply x+8 by x^{2}.

$$\frac{1}{x^{3}+8x^{2}}$$

Differentiate w.r.t. x

$$\frac{-3x-16}{\left(x+8\right)^{2}x^{3}}$$

Steps Using Chain Rule

Express \frac{\frac{x}{x+8}}{x^{3}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{\left(x+8\right)x^{3}})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\left(x+8\right)x^{2}})$$

Use the distributive property to multiply x+8 by x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{3}+8x^{2}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(x^{3}+8x^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{3}+8x^{2})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\left(x^{3}+8x^{2}\right)^{-2}\left(3x^{3-1}+2\times 8x^{2-1}\right)$$

Simplify.

$$\left(x^{3}+8x^{2}\right)^{-2}\left(-3x^{2}-16x^{1}\right)$$

For any term t, t^{1}=t.

$$\left(x^{3}+8x^{2}\right)^{-2}\left(-3x^{2}-16x\right)$$