Solve d/dx(5x^2/4-8/3x^2)

$$\frac { d } { d x } ( \frac { 5 x ^ { 2 } } { 4 } - \frac { 8 } { 3 x ^ { 2 } } )$$

Evaluate

$$\frac{5x}{2}+\frac{16}{3x^{3}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 3x^{2} is 12x^{2}. Multiply \frac{5x^{2}}{4} times \frac{3x^{2}}{3x^{2}}. Multiply \frac{8}{3x^{2}} times \frac{4}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{2}\times 3x^{2}}{12x^{2}}-\frac{8\times 4}{12x^{2}})$$

Since \frac{5x^{2}\times 3x^{2}}{12x^{2}} and \frac{8\times 4}{12x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x^{2}\times 3x^{2}-8\times 4}{12x^{2}})$$

Do the multiplications in 5x^{2}\times 3x^{2}-8\times 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{15x^{4}-32}{12x^{2}})$$

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.

$$\frac{12x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(15x^{4}-32)-\left(15x^{4}-32\right)\frac{\mathrm{d}}{\mathrm{d}x}(12x^{2})}{\left(12x^{2}\right)^{2}}$$

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}.

$$\frac{12x^{2}\times 4\times 15x^{4-1}-\left(15x^{4}-32\right)\times 2\times 12x^{2-1}}{\left(12x^{2}\right)^{2}}$$

Do the arithmetic.

$$\frac{12x^{2}\times 60x^{3}-\left(15x^{4}-32\right)\times 24x^{1}}{\left(12x^{2}\right)^{2}}$$

Expand using distributive property.

$$\frac{12x^{2}\times 60x^{3}-\left(15x^{4}\times 24x^{1}-32\times 24x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

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

$$\frac{12\times 60x^{2+3}-\left(15\times 24x^{4+1}-32\times 24x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

Do the arithmetic.

$$\frac{720x^{5}-\left(360x^{5}-768x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{720x^{5}-360x^{5}-\left(-768x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

Combine like terms.

$$\frac{\left(720-360\right)x^{5}-\left(-768x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

Subtract 360 from 720.

$$\frac{360x^{5}-\left(-768x^{1}\right)}{\left(12x^{2}\right)^{2}}$$

Factor out 24x.

$$\frac{24x\left(15x^{4}-\left(-32x^{0}\right)\right)}{\left(12x^{2}\right)^{2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\frac{24x\left(15x^{4}-\left(-32x^{0}\right)\right)}{12^{2}\left(x^{2}\right)^{2}}$$

Raise 12 to the power 2.

$$\frac{24x\left(15x^{4}-\left(-32x^{0}\right)\right)}{144\left(x^{2}\right)^{2}}$$

To raise a power to another power, multiply the exponents.

$$\frac{24x\left(15x^{4}-\left(-32x^{0}\right)\right)}{144x^{2\times 2}}$$

Multiply 2 times 2.

$$\frac{24x\left(15x^{4}-\left(-32x^{0}\right)\right)}{144x^{4}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{24\left(15x^{4}-\left(-32x^{0}\right)\right)}{144x^{4-1}}$$

Subtract 1 from 4.

$$\frac{24\left(15x^{4}-\left(-32x^{0}\right)\right)}{144x^{3}}$$

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

$$\frac{24\left(15x^{4}-\left(-32\right)\right)}{144x^{3}}$$

Differentiate w.r.t. x

$$\frac{5}{2}-\frac{16}{x^{4}}$$