Solve left(5x+3yright)^255x*1sqrt{x^2}

$${ \left(5x+3y \right) }^{ 2 } 55x \times 1 \sqrt{ { x }^{ 2 } }$$

Differentiate w.r.t. x

$$110|x|\left(5x+3y\right)\left(10x+3y\right)$$

Steps Using Derivative Rule for Sum

Multiply 1 and 55 to get 55.

$$\frac{\mathrm{d}}{\mathrm{d}x}(55\left(5x+3y\right)^{2}x\sqrt{x^{2}})$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+3y\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(55\left(25x^{2}+30xy+9y^{2}\right)x\sqrt{x^{2}})$$

Use the distributive property to multiply 55 by 25x^{2}+30xy+9y^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1375x^{2}+1650xy+495y^{2}\right)x\sqrt{x^{2}})$$

Use the distributive property to multiply 1375x^{2}+1650xy+495y^{2} by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1375x^{3}+1650yx^{2}+495y^{2}x\right)\sqrt{x^{2}})$$

Use the distributive property to multiply 1375x^{3}+1650yx^{2}+495y^{2}x by \sqrt{x^{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(1375x^{3}\sqrt{x^{2}}+1650yx^{2}\sqrt{x^{2}}+495y^{2}x\sqrt{x^{2}})$$

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}.

$$3\times 1375\sqrt{x^{2}}x^{3-1}+2\times 1650y\sqrt{x^{2}}x^{2-1}+495\sqrt{x^{2}}y^{2}x^{1-1}$$

Multiply 3 times 1375\sqrt{x^{2}}.

$$4125|x|x^{3-1}+2\times 1650y\sqrt{x^{2}}x^{2-1}+495\sqrt{x^{2}}y^{2}x^{1-1}$$

Subtract 1 from 3.

$$4125|x|x^{2}+2\times 1650y\sqrt{x^{2}}x^{2-1}+495\sqrt{x^{2}}y^{2}x^{1-1}$$

Multiply 2 times 1650y\sqrt{x^{2}}.

$$4125|x|x^{2}+3300y|x|x^{2-1}+495\sqrt{x^{2}}y^{2}x^{1-1}$$

Subtract 1 from 2.

$$4125|x|x^{2}+3300y|x|x^{1}+495\sqrt{x^{2}}y^{2}x^{1-1}$$

Subtract 1 from 1.

$$4125|x|x^{2}+3300y|x|x^{1}+495|x|y^{2}x^{0}$$

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

$$4125|x|x^{2}+3300y|x|x+495|x|y^{2}x^{0}$$

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

$$4125|x|x^{2}+3300y|x|x+495|x|y^{2}\times 1$$

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

$$4125|x|x^{2}+3300y|x|x+495|x|y^{2}$$

Evaluate

$$55x|x|\left(5x+3y\right)^{2}$$