Solve 4c/16c^2-9

$$\frac { 4 c } { 16 c ^ { 2 } - 9 }$$

Differentiate w.r.t. c

$$-\frac{4\left(16c^{2}+9\right)}{\left(16c^{2}-9\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

$$\frac{\left(16c^{2}-9\right)\frac{\mathrm{d}}{\mathrm{d}c}(4c^{1})-4c^{1}\frac{\mathrm{d}}{\mathrm{d}c}(16c^{2}-9)}{\left(16c^{2}-9\right)^{2}}$$

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}.

$$\frac{\left(16c^{2}-9\right)\times 4c^{1-1}-4c^{1}\times 2\times 16c^{2-1}}{\left(16c^{2}-9\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(16c^{2}-9\right)\times 4c^{0}-4c^{1}\times 32c^{1}}{\left(16c^{2}-9\right)^{2}}$$

Expand using distributive property.

$$\frac{16c^{2}\times 4c^{0}-9\times 4c^{0}-4c^{1}\times 32c^{1}}{\left(16c^{2}-9\right)^{2}}$$

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

$$\frac{16\times 4c^{2}-9\times 4c^{0}-4\times 32c^{1+1}}{\left(16c^{2}-9\right)^{2}}$$

Do the arithmetic.

$$\frac{64c^{2}-36c^{0}-128c^{2}}{\left(16c^{2}-9\right)^{2}}$$

Combine like terms.

$$\frac{\left(64-128\right)c^{2}-36c^{0}}{\left(16c^{2}-9\right)^{2}}$$

Subtract 128 from 64.

$$\frac{-64c^{2}-36c^{0}}{\left(16c^{2}-9\right)^{2}}$$

Factor out 4.

$$\frac{4\left(-16c^{2}-9c^{0}\right)}{\left(16c^{2}-9\right)^{2}}$$

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

$$\frac{4\left(-16c^{2}-9\right)}{\left(16c^{2}-9\right)^{2}}$$

Evaluate

$$\frac{4c}{16c^{2}-9}$$