Solve x^2*2sqrt{2}*sqrt{x*1}+3x*5sqrt{2}*sqrt{x^3}

$${ x }^{ 2 } \times 2 \sqrt{ 2 } \times \sqrt{ x \times 1 } +3x \times 5 \sqrt{ 2 } \times \sqrt{ { x }^{ 3 } }$$

Differentiate w.r.t. x

$$\frac{85\sqrt{2}x^{\frac{3}{2}}}{2}$$

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}.

$$2\times 2\sqrt{2}\sqrt{x}x^{2-1}+15\sqrt{2}\sqrt{x^{3}}x^{1-1}$$

Multiply 2 times 2\sqrt{2}\sqrt{x}.

$$4\sqrt{2x}x^{2-1}+15\sqrt{2}\sqrt{x^{3}}x^{1-1}$$

Subtract 1 from 2.

$$4\sqrt{2x}x^{1}+15\sqrt{2}\sqrt{x^{3}}x^{1-1}$$

Subtract 1 from 1.

$$4\sqrt{2x}x^{1}+\frac{15\times \left(2x\right)^{\frac{3}{2}}}{2}x^{0}$$

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

$$4\sqrt{2x}x+\frac{15\times \left(2x\right)^{\frac{3}{2}}}{2}x^{0}$$

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

$$4\sqrt{2x}x+\frac{15\times \left(2x\right)^{\frac{3}{2}}}{2}\times 1$$

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

$$4\sqrt{2x}x+\frac{15\times \left(2x\right)^{\frac{3}{2}}}{2}$$

Evaluate

$$17\sqrt{2}x^{\frac{5}{2}}$$