Solve -1/2x*(-4x^3)-(-3x^2)*(1/3x^2)

$$- \frac { 1 } { 2 } x \cdot ( - 4 x ^ { 3 } ) - ( - 3 x ^ { 2 } ) \cdot ( \frac { 1 } { 3 } x ^ { 2 } )$$

Evaluate

$$3x^{4}$$

Solution Steps

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

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

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Multiply -\frac{1}{2} and -4 to get 2.

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

Multiply -3 and \frac{1}{3} to get -1.

$$2x^{4}-\left(-x^{4}\right)$$

The opposite of -x^{4} is x^{4}.

$$2x^{4}+x^{4}$$

Combine 2x^{4} and x^{4} to get 3x^{4}.

$$3x^{4}$$

Differentiate w.r.t. x

$$12x^{3}$$

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}.

$$2x^{3}x^{1-1}+2x^{2}x^{2-1}$$

Subtract 1 from 1.

$$2x^{3}x^{0}+2x^{2}x^{2-1}$$

Subtract 1 from 2.

$$2x^{3}x^{0}+2x^{2}x^{1}$$

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

$$2x^{3}x^{0}+2x^{2}x$$

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

$$2x^{3}\times 1+2x^{2}x$$

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

$$2x^{3}+2x^{2}x$$