Solve (y^{2})^3*y^2-2(y^2)^4

$$( y ^ { 2 } ) ^ { 3 } \cdot y ^ { 2 } - 2 ( y ^ { 2 } ) ^ { 4 }$$

Evaluate

$$-y^{8}$$

Solution Steps

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

$$y^{6}y^{2}-2\left(y^{2}\right)^{4}$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$y^{8}-2\left(y^{2}\right)^{4}$$

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

$$y^{8}-2y^{8}$$

Combine y^{8} and -2y^{8} to get -y^{8}.

$$-y^{8}$$

Differentiate w.r.t. y

$$-8y^{7}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{6}y^{2}-2\left(y^{2}\right)^{4})$$

To multiply powers of the same base, add their exponents. Add 6 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{8}-2\left(y^{2}\right)^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{8}-2y^{8})$$

Combine y^{8} and -2y^{8} to get -y^{8}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(-y^{8})$$

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

$$8\left(-1\right)y^{8-1}$$

Multiply 8 times -1.

$$-8y^{8-1}$$

Subtract 1 from 8.

$$-8y^{7}$$