Solve (-5xy)*3x^2y-12x^3*(-7/4y^2)

$$( - 5 x y ) \cdot 3 x ^ { 2 } y - 12 x ^ { 3 } \cdot ( - \frac { 7 } { 4 } y ^ { 2 } )$$

Evaluate

$$6y^{2}x^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-5x^{3}y\times 3y-12x^{3}\left(-\frac{7}{4}\right)y^{2}$$

Multiply y and y to get y^{2}.

$$-5x^{3}y^{2}\times 3-12x^{3}\left(-\frac{7}{4}\right)y^{2}$$

Multiply -5 and 3 to get -15.

$$-15x^{3}y^{2}-12x^{3}\left(-\frac{7}{4}\right)y^{2}$$

Multiply 12 and -\frac{7}{4} to get -21.

$$-15x^{3}y^{2}-\left(-21x^{3}y^{2}\right)$$

The opposite of -21x^{3}y^{2} is 21x^{3}y^{2}.

$$-15x^{3}y^{2}+21x^{3}y^{2}$$

Combine -15x^{3}y^{2} and 21x^{3}y^{2} to get 6x^{3}y^{2}.

$$6x^{3}y^{2}$$

Differentiate w.r.t. x

$$18\left(xy\right)^{2}$$

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}.

$$\left(-15x^{2}y^{2}\right)x^{1-1}+3\times 21y^{2}x^{3-1}$$

Subtract 1 from 1.

$$\left(-15x^{2}y^{2}\right)x^{0}+3\times 21y^{2}x^{3-1}$$

Multiply 3 times -12\left(-\frac{7}{4}\right)y^{2}.

$$\left(-15x^{2}y^{2}\right)x^{0}+63y^{2}x^{3-1}$$

Subtract 1 from 3.

$$\left(-15x^{2}y^{2}\right)x^{0}+63y^{2}x^{2}$$

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

$$\left(-15x^{2}y^{2}\right)\times 1+63y^{2}x^{2}$$

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

$$-15x^{2}y^{2}+63y^{2}x^{2}$$