Solve 30/1+16k^2-1

$$\frac { 30 } { 1 + 16 k ^ { 2 } } - 1$$

Evaluate

$$\frac{29-16k^{2}}{16k^{2}+1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+16k^{2}}{1+16k^{2}}.

$$\frac{30}{1+16k^{2}}-\frac{1+16k^{2}}{1+16k^{2}}$$

Since \frac{30}{1+16k^{2}} and \frac{1+16k^{2}}{1+16k^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{30-\left(1+16k^{2}\right)}{1+16k^{2}}$$

Do the multiplications in 30-\left(1+16k^{2}\right).

$$\frac{30-1-16k^{2}}{1+16k^{2}}$$

Combine like terms in 30-1-16k^{2}.

$$\frac{29-16k^{2}}{1+16k^{2}}$$

Differentiate w.r.t. k

$$-\frac{960k}{\left(16k^{2}+1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+16k^{2}}{1+16k^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{30}{1+16k^{2}}-\frac{1+16k^{2}}{1+16k^{2}})$$

Since \frac{30}{1+16k^{2}} and \frac{1+16k^{2}}{1+16k^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{30-\left(1+16k^{2}\right)}{1+16k^{2}})$$

Do the multiplications in 30-\left(1+16k^{2}\right).

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{30-1-16k^{2}}{1+16k^{2}})$$

Combine like terms in 30-1-16k^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{29-16k^{2}}{1+16k^{2}})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(16k^{2}+1\right)\frac{\mathrm{d}}{\mathrm{d}k}(-16k^{2}+29)-\left(-16k^{2}+29\right)\frac{\mathrm{d}}{\mathrm{d}k}(16k^{2}+1)}{\left(16k^{2}+1\right)^{2}}$$

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}.

$$\frac{\left(16k^{2}+1\right)\times 2\left(-16\right)k^{2-1}-\left(-16k^{2}+29\right)\times 2\times 16k^{2-1}}{\left(16k^{2}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(16k^{2}+1\right)\left(-32\right)k^{1}-\left(-16k^{2}+29\right)\times 32k^{1}}{\left(16k^{2}+1\right)^{2}}$$

Expand using distributive property.

$$\frac{16k^{2}\left(-32\right)k^{1}-32k^{1}-\left(-16k^{2}\times 32k^{1}+29\times 32k^{1}\right)}{\left(16k^{2}+1\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{16\left(-32\right)k^{2+1}-32k^{1}-\left(-16\times 32k^{2+1}+29\times 32k^{1}\right)}{\left(16k^{2}+1\right)^{2}}$$

Do the arithmetic.

$$\frac{-512k^{3}-32k^{1}-\left(-512k^{3}+928k^{1}\right)}{\left(16k^{2}+1\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{-512k^{3}-32k^{1}-\left(-512k^{3}\right)-928k^{1}}{\left(16k^{2}+1\right)^{2}}$$

Combine like terms.

$$\frac{\left(-512-\left(-512\right)\right)k^{3}+\left(-32-928\right)k^{1}}{\left(16k^{2}+1\right)^{2}}$$

Subtract -512 from -512 and 928 from -32.

$$\frac{-960k^{1}}{\left(16k^{2}+1\right)^{2}}$$

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

$$\frac{-960k}{\left(16k^{2}+1\right)^{2}}$$