Solve (1/20)/(400-x^2)

$$( 1 / 20 ) / ( 400 - x ^ { 2 } )$$

Evaluate

$$\frac{1}{20\left(400-x^{2}\right)}$$

Solution Steps

Express \frac{\frac{1}{20}}{400-x^{2}} as a single fraction.

$$\frac{1}{20\left(400-x^{2}\right)}$$

Use the distributive property to multiply 20 by 400-x^{2}.

$$\frac{1}{8000-20x^{2}}$$

Differentiate w.r.t. x

$$\frac{x}{10\left(400-x^{2}\right)^{2}}$$

Steps Using Chain Rule

Express \frac{\frac{1}{20}}{400-x^{2}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{20\left(400-x^{2}\right)})$$

Use the distributive property to multiply 20 by 400-x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{8000-20x^{2}})$$

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).

$$-\left(-20x^{2}+8000\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(-20x^{2}+8000)$$

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}.

$$-\left(-20x^{2}+8000\right)^{-2}\times 2\left(-20\right)x^{2-1}$$

Simplify.

$$40x^{1}\left(-20x^{2}+8000\right)^{-2}$$

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

$$40x\left(-20x^{2}+8000\right)^{-2}$$