Solve 5x^{9}sqrt{3x}+2x^2sqrt{12x^15}+4x^8sqrt{27x^3}

$$5 x ^ { 9 } \sqrt { 3 x } + 2 x ^ { 2 } \sqrt { 12 x ^ { 15 } } + 4 x ^ { 8 } \sqrt { 27 x ^ { 3 } }$$

Differentiate w.r.t. x

$$\frac{399\sqrt{3}x^{\frac{17}{2}}}{2}$$

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}.

$$9\times 5\sqrt{3x}x^{9-1}+2\times 2\sqrt{12x^{15}}x^{2-1}+8\times 4\sqrt{27x^{3}}x^{8-1}$$

Multiply 9 times 5\sqrt{3x}.

$$45\sqrt{3x}x^{9-1}+2\times 2\sqrt{12x^{15}}x^{2-1}+8\times 4\sqrt{27x^{3}}x^{8-1}$$

Subtract 1 from 9.

$$45\sqrt{3x}x^{8}+2\times 2\sqrt{12x^{15}}x^{2-1}+8\times 4\sqrt{27x^{3}}x^{8-1}$$

Multiply 2 times 2\sqrt{12x^{15}}.

$$45\sqrt{3x}x^{8}+\frac{8\times \left(3x\right)^{\frac{15}{2}}}{2187}x^{2-1}+8\times 4\sqrt{27x^{3}}x^{8-1}$$

Subtract 1 from 2.

$$45\sqrt{3x}x^{8}+\frac{8\times \left(3x\right)^{\frac{15}{2}}}{2187}x^{1}+8\times 4\sqrt{27x^{3}}x^{8-1}$$

Multiply 2 times 2\sqrt{12x^{15}}.

$$45\sqrt{3x}x^{8}+\frac{8\times \left(3x\right)^{\frac{15}{2}}}{2187}x^{1}+32\times \left(3x\right)^{\frac{3}{2}}x^{8-1}$$

Subtract 1 from 8.

$$45\sqrt{3x}x^{8}+\frac{8\times \left(3x\right)^{\frac{15}{2}}}{2187}x^{1}+32\times \left(3x\right)^{\frac{3}{2}}x^{7}$$

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

$$45\sqrt{3x}x^{8}+\frac{8\times \left(3x\right)^{\frac{15}{2}}}{2187}x+32\times \left(3x\right)^{\frac{3}{2}}x^{7}$$

Evaluate

$$21\sqrt{3}x^{\frac{19}{2}}$$