Solve 8*0.5^3/x^2+4*0.5^2

$$\frac{ 8 \times { 0.5 }^{ 3 } }{ { x }^{ 2 } +4 \times { 0.5 }^{ 2 } }$$

Evaluate

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

Solution Steps

Calculate 0.5 to the power of 3 and get 0.125.

$$\frac{8\times 0.125}{x^{2}+4\times 0.5^{2}}$$

Multiply 8 and 0.125 to get 1.

$$\frac{1}{x^{2}+4\times 0.5^{2}}$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\frac{1}{x^{2}+4\times 0.25}$$

Multiply 4 and 0.25 to get 1.

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Calculate 0.5 to the power of 3 and get 0.125.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8\times 0.125}{x^{2}+4\times 0.5^{2}})$$

Multiply 8 and 0.125 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{2}+4\times 0.5^{2}})$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{2}+4\times 0.25})$$

Multiply 4 and 0.25 to get 1.

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

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).

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

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}.

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

Simplify.

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

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

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