Solve 5x^3(1^1/3-x^2/3

$$5 x ^ { 3 } ( 1 ^ { 1 / 3 } - x ^ { 2 / 3 }$$

Evaluate

$$5x^{3}-5x^{\frac{11}{3}}$$

Solution Steps

Calculate 1 to the power of \frac{1}{3} and get 1.

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

Use the distributive property to multiply 5x^{3} by 1-x^{\frac{2}{3}}.

$$5x^{3}-5x^{\frac{11}{3}}$$

Differentiate w.r.t. x

$$-\frac{55x^{\frac{8}{3}}}{3}+15x^{2}$$

Steps Using Derivative Rule for Sum

Calculate 1 to the power of \frac{1}{3} and get 1.

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

Use the distributive property to multiply 5x^{3} by 1-x^{\frac{2}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{3}-5x^{\frac{11}{3}})$$

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}.

$$3\times 5x^{3-1}+\frac{11}{3}\left(-5\right)x^{\frac{11}{3}-1}$$

Multiply 3 times 5.

$$15x^{3-1}+\frac{11}{3}\left(-5\right)x^{\frac{11}{3}-1}$$

Subtract 1 from 3.

$$15x^{2}+\frac{11}{3}\left(-5\right)x^{\frac{11}{3}-1}$$

Multiply \frac{11}{3} times -5.

$$15x^{2}-\frac{55}{3}x^{\frac{11}{3}-1}$$

Subtract 1 from \frac{11}{3}.

$$15x^{2}-\frac{55}{3}x^{\frac{8}{3}}$$