Solve (7x^3)^3(x^9)^frac{1{2}}

$$( 7 x ^ { 3 } ) ^ { 3 } ( x ^ { 9 } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$343x^{\frac{27}{2}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{2} to get \frac{9}{2}.

$$\left(7x^{3}\right)^{3}x^{\frac{9}{2}}$$

Expand \left(7x^{3}\right)^{3}.

$$7^{3}\left(x^{3}\right)^{3}x^{\frac{9}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$7^{3}x^{9}x^{\frac{9}{2}}$$

Calculate 7 to the power of 3 and get 343.

$$343x^{9}x^{\frac{9}{2}}$$

To multiply powers of the same base, add their exponents. Add 9 and \frac{9}{2} to get \frac{27}{2}.

$$343x^{\frac{27}{2}}$$

Differentiate w.r.t. x

$$\frac{9261x^{\frac{25}{2}}}{2}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 9 and \frac{1}{2} to get \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(7x^{3}\right)^{3}x^{\frac{9}{2}})$$

Expand \left(7x^{3}\right)^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7^{3}\left(x^{3}\right)^{3}x^{\frac{9}{2}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7^{3}x^{9}x^{\frac{9}{2}})$$

Calculate 7 to the power of 3 and get 343.

$$\frac{\mathrm{d}}{\mathrm{d}x}(343x^{9}x^{\frac{9}{2}})$$

To multiply powers of the same base, add their exponents. Add 9 and \frac{9}{2} to get \frac{27}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(343x^{\frac{27}{2}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{27}{2}\times 343x^{\frac{27}{2}-1}$$

Multiply \frac{27}{2} times 343.

$$\frac{9261}{2}x^{\frac{27}{2}-1}$$

Subtract 1 from \frac{27}{2}.

$$\frac{9261}{2}x^{\frac{25}{2}}$$