Solve 10^-2/2*10^-2*x+1+x^2*10^-7

$$\frac{ { 10 }^{ -2 } }{ 2 \times { 10 }^{ -2 } \times x+1+ { x }^{ 2 } \times { 10 }^{ -7 } }$$

Evaluate

$$\frac{100000}{x^{2}+200000x+10000000}$$

Solution Steps

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\frac{1}{100}}{2\times 10^{-2}x+1+x^{2}\times 10^{-7}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\frac{1}{100}}{2\times \frac{1}{100}x+1+x^{2}\times 10^{-7}}$$

Multiply 2 and \frac{1}{100} to get \frac{1}{50}.

$$\frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times 10^{-7}}$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$\frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}}$$

Express \frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}} as a single fraction.

$$\frac{1}{100\left(\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}\right)}$$

Use the distributive property to multiply 100 by \frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}.

$$\frac{1}{2x+100+100x^{2}\times \frac{1}{10000000}}$$

Multiply 100 and \frac{1}{10000000} to get \frac{1}{100000}.

$$\frac{1}{2x+100+\frac{1}{100000}x^{2}}$$

Differentiate w.r.t. x

$$-\frac{200000\left(x+100000\right)}{\left(x^{2}+200000x+10000000\right)^{2}}$$

Steps Using Chain Rule

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{100}}{2\times 10^{-2}x+1+x^{2}\times 10^{-7}})$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{100}}{2\times \frac{1}{100}x+1+x^{2}\times 10^{-7}})$$

Multiply 2 and \frac{1}{100} to get \frac{1}{50}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times 10^{-7}})$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}})$$

Express \frac{\frac{1}{100}}{\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{100\left(\frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}\right)})$$

Use the distributive property to multiply 100 by \frac{1}{50}x+1+x^{2}\times \frac{1}{10000000}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2x+100+100x^{2}\times \frac{1}{10000000}})$$

Multiply 100 and \frac{1}{10000000} to get \frac{1}{100000}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2x+100+\frac{1}{100000}x^{2}})$$

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).

$$-\left(2x^{1}+\frac{1}{100000}x^{2}+100\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(2x^{1}+\frac{1}{100000}x^{2}+100)$$

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}.

$$-\left(2x^{1}+\frac{1}{100000}x^{2}+100\right)^{-2}\left(2x^{1-1}+2\times \frac{1}{100000}x^{2-1}\right)$$

Simplify.

$$\left(2x^{1}+\frac{1}{100000}x^{2}+100\right)^{-2}\left(-2x^{0}-\frac{1}{50000}x^{1}\right)$$

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

$$\left(2x+\frac{1}{100000}x^{2}+100\right)^{-2}\left(-2x^{0}-\frac{1}{50000}x\right)$$

For any term t except 0, t^{0}=1.

$$\left(2x+\frac{1}{100000}x^{2}+100\right)^{-2}\left(-2-\frac{1}{50000}x\right)$$