Solve 33x/4x^5-3x-4

$$\frac{ 33x }{ 4 { x }^{ 5 } -3x-4 }$$

Differentiate w.r.t. x

$$-\frac{132\left(4x^{5}+1\right)}{\left(4x^{5}-3x-4\right)^{2}}$$

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.

$$\frac{\left(4x^{5}-3x^{1}-4\right)\frac{\mathrm{d}}{\mathrm{d}x}(33x^{1})-33x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(4x^{5}-3x^{1}-4)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

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}.

$$\frac{\left(4x^{5}-3x^{1}-4\right)\times 33x^{1-1}-33x^{1}\left(5\times 4x^{5-1}-3x^{1-1}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

Simplify.

$$\frac{\left(4x^{5}-3x^{1}-4\right)\times 33x^{0}-33x^{1}\left(20x^{4}-3x^{0}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

Multiply 4x^{5}-3x^{1}-4 times 33x^{0}.

$$\frac{4x^{5}\times 33x^{0}-3x^{1}\times 33x^{0}-4\times 33x^{0}-33x^{1}\left(20x^{4}-3x^{0}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

Multiply 33x^{1} times 20x^{4}-3x^{0}.

$$\frac{4x^{5}\times 33x^{0}-3x^{1}\times 33x^{0}-4\times 33x^{0}-\left(33x^{1}\times 20x^{4}+33x^{1}\left(-3\right)x^{0}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

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

$$\frac{4\times 33x^{5}-3\times 33x^{1}-4\times 33x^{0}-\left(33\times 20x^{1+4}+33\left(-3\right)x^{1}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

Simplify.

$$\frac{132x^{5}-99x^{1}-132x^{0}-\left(660x^{5}-99x^{1}\right)}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

Combine like terms.

$$\frac{-528x^{5}-9x^{4}-132x^{0}}{\left(4x^{5}-3x^{1}-4\right)^{2}}$$

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

$$\frac{-528x^{5}-9x^{4}-132x^{0}}{\left(4x^{5}-3x-4\right)^{2}}$$

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

$$\frac{-528x^{5}-9x^{4}-132}{\left(4x^{5}-3x-4\right)^{2}}$$

Evaluate

$$\frac{33x}{4x^{5}-3x-4}$$