Solve (-2/3w^4)+(-w^3)*(-9/10w)

$$( - \frac { 2 } { 3 } w ^ { 4 } ) + ( - w ^ { 3 } ) \times ( - \frac { 9 } { 10 } w )$$

Evaluate

$$\frac{7w^{4}}{30}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$-\frac{2}{3}w^{4}-w^{4}\left(-\frac{9}{10}\right)$$

Multiply -1 and -\frac{9}{10} to get \frac{9}{10}.

$$-\frac{2}{3}w^{4}+\frac{9}{10}w^{4}$$

Combine -\frac{2}{3}w^{4} and \frac{9}{10}w^{4} to get \frac{7}{30}w^{4}.

$$\frac{7}{30}w^{4}$$

Differentiate w.r.t. w

$$\frac{14w^{3}}{15}$$

Steps Using Derivative Rule for Sum

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}.

$$4\left(-\frac{2}{3}\right)w^{4-1}+\left(-\left(-\frac{9w^{3}}{10}\right)\right)w^{1-1}$$

Multiply 4 times -\frac{2}{3}.

$$-\frac{8}{3}w^{4-1}+\left(-\left(-\frac{9w^{3}}{10}\right)\right)w^{1-1}$$

Subtract 1 from 4.

$$-\frac{8}{3}w^{3}+\left(-\left(-\frac{9w^{3}}{10}\right)\right)w^{1-1}$$

Subtract 1 from 1.

$$-\frac{8}{3}w^{3}+\frac{9w^{3}}{10}w^{0}$$

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

$$-\frac{8}{3}w^{3}+\frac{9w^{3}}{10}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$-\frac{8}{3}w^{3}+\frac{9w^{3}}{10}$$