Solve 2(x^{3})^{4}+x^{4}*(x^{4})^{2}+x^{5}*x^7+x^6*(x^3)^2

$$2 ( x ^ { 3 } ) ^ { 4 } + x ^ { 4 } \cdot ( x ^ { 4 } ) ^ { 2 } + x ^ { 5 } \cdot x ^ { 7 } + x ^ { 6 } \cdot ( x ^ { 3 } ) ^ { 2 }$$

Evaluate

$$5x^{12}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$2x^{12}+x^{4}\left(x^{4}\right)^{2}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$2x^{12}+x^{4}x^{8}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$2x^{12}+x^{12}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 5 and 7 to get 12.

$$2x^{12}+x^{12}+x^{12}+x^{6}\left(x^{3}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$2x^{12}+x^{12}+x^{12}+x^{6}x^{6}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$2x^{12}+x^{12}+x^{12}+x^{12}$$

Combine 2x^{12} and x^{12} to get 3x^{12}.

$$3x^{12}+x^{12}+x^{12}$$

Combine 3x^{12} and x^{12} to get 4x^{12}.

$$4x^{12}+x^{12}$$

Combine 4x^{12} and x^{12} to get 5x^{12}.

$$5x^{12}$$

Differentiate w.r.t. x

$$60x^{11}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{4}\left(x^{4}\right)^{2}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{4}x^{8}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2})$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{12}+x^{5}x^{7}+x^{6}\left(x^{3}\right)^{2})$$

To multiply powers of the same base, add their exponents. Add 5 and 7 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{12}+x^{12}+x^{6}\left(x^{3}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{12}+x^{12}+x^{6}x^{6})$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{12}+x^{12}+x^{12}+x^{12})$$

Combine 2x^{12} and x^{12} to get 3x^{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{12}+x^{12}+x^{12})$$

Combine 3x^{12} and x^{12} to get 4x^{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{12}+x^{12})$$

Combine 4x^{12} and x^{12} to get 5x^{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{12})$$

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

$$12\times 5x^{12-1}$$

Multiply 12 times 5.

$$60x^{12-1}$$

Subtract 1 from 12.

$$60x^{11}$$