Solve 4x^3/3x^3+7x

$$\frac { 4 x ^ { 3 } } { 3 x ^ { 3 } + 7 x }$$

Evaluate

$$\frac{4x^{2}}{3x^{2}+7}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{4x^{3}}{x\left(3x^{2}+7\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{4x^{2}}{3x^{2}+7}$$

Differentiate w.r.t. x

$$\frac{56x}{\left(3x^{2}+7\right)^{2}}$$

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.

$$\frac{\left(3x^{3}+7x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3})-4x^{3}\frac{\mathrm{d}}{\mathrm{d}x}(3x^{3}+7x^{1})}{\left(3x^{3}+7x^{1}\right)^{2}}$$

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}.

$$\frac{\left(3x^{3}+7x^{1}\right)\times 3\times 4x^{3-1}-4x^{3}\left(3\times 3x^{3-1}+7x^{1-1}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

Simplify.

$$\frac{\left(3x^{3}+7x^{1}\right)\times 12x^{2}-4x^{3}\left(9x^{2}+7x^{0}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

Multiply 3x^{3}+7x^{1} times 12x^{2}.

$$\frac{3x^{3}\times 12x^{2}+7x^{1}\times 12x^{2}-4x^{3}\left(9x^{2}+7x^{0}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

Multiply 4x^{3} times 9x^{2}+7x^{0}.

$$\frac{3x^{3}\times 12x^{2}+7x^{1}\times 12x^{2}-\left(4x^{3}\times 9x^{2}+4x^{3}\times 7x^{0}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

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

$$\frac{3\times 12x^{3+2}+7\times 12x^{1+2}-\left(4\times 9x^{3+2}+4\times 7x^{3}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

Simplify.

$$\frac{36x^{5}+84x^{3}-\left(36x^{5}+28x^{3}\right)}{\left(3x^{3}+7x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{56x^{3}}{\left(3x^{3}+7x^{1}\right)^{2}}$$

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

$$\frac{56x^{3}}{\left(3x^{3}+7x\right)^{2}}$$