Solve 1/2(5x^{2}+1)^-frac{1{2}}*10x*(3x^2+2)+15x^2+1)^frac{1{2}}*6x

$$\frac { 1 } { 2 } ( 5 x ^ { 2 } + 1 ) ^ { - \frac { 1 } { 2 } } \cdot 10 x \cdot ( 3 x ^ { 2 } + 2 ) + 15 x ^ { 2 } + 1 ) ^ { \frac { 1 } { 2 } } \cdot 6 x$$

Differentiate w.r.t. x

$$\frac{5\left(6x\left(5x^{2}+1\right)^{\frac{3}{2}}+30x^{4}+9x^{2}+2\right)}{\left(5x^{2}+1\right)^{\frac{3}{2}}}$$

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}.

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}x^{1-1}+2\times 15x^{2-1}$$

Subtract 1 from 1.

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}x^{0}+2\times 15x^{2-1}$$

Multiply 2 times 15.

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}x^{0}+30x^{2-1}$$

Subtract 1 from 2.

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}x^{0}+30x^{1}$$

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

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}x^{0}+30x$$

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

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}\times 1+30x$$

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

$$\frac{5\left(3x^{2}+2\right)}{\sqrt{5x^{2}+1}}+30x$$

Evaluate

$$\frac{15x^{3}}{\sqrt{5x^{2}+1}}+15x^{2}+\frac{10x}{\sqrt{5x^{2}+1}}+1$$