Solve (x^{3/4})^{2}*(6sqrt{x^5})^3+(x^frac{3{7}})^-7

$$( x ^ { \frac { 3 } { 4 } } ) ^ { 2 } \times ( 6 \sqrt { x ^ { 5 } } ) ^ { 3 } + ( x ^ { \frac { 3 } { 7 } } ) ^ { - 7 }$$

Differentiate w.r.t. x

$$1944x^{8}-\frac{3}{x^{4}}$$

Steps Using Derivative Rule for Sum

To raise a power to another power, multiply the exponents. Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{2}}\times \left(6\sqrt{x^{5}}\right)^{3}+\left(x^{\frac{3}{7}}\right)^{-7})$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{7} and -7 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{2}}\times \left(6\sqrt{x^{5}}\right)^{3}+x^{-3})$$

Expand \left(6\sqrt{x^{5}}\right)^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{2}}\times 6^{3}\left(\sqrt{x^{5}}\right)^{3}+x^{-3})$$

Calculate 6 to the power of 3 and get 216.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{3}{2}}\times 216\left(\sqrt{x^{5}}\right)^{3}+x^{-3})$$

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}.

$$\frac{3}{2}\times 216\left(\sqrt{x^{5}}\right)^{3}x^{\frac{3}{2}-1}-3x^{-3-1}$$

Multiply \frac{3}{2} times 216\left(\sqrt{x^{5}}\right)^{3}.

$$324x^{\frac{15}{2}}x^{\frac{3}{2}-1}-3x^{-3-1}$$

Subtract 1 from \frac{3}{2}.

$$324x^{\frac{15}{2}}\sqrt{x}-3x^{-3-1}$$

Subtract 1 from -3.

$$324x^{\frac{15}{2}}\sqrt{x}-3x^{-4}$$

Evaluate

$$216x^{9}+\frac{1}{x^{3}}$$