Solve 2y^3*3y^5+(y^2)^4

$$2 y ^ { 3 } \times 3 y ^ { 5 } + ( y ^ { 2 } ) ^ { 4 }$$

Evaluate

$$7y^{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$2y^{8}\times 3+\left(y^{2}\right)^{4}$$

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

$$2y^{8}\times 3+y^{8}$$

Multiply 2 and 3 to get 6.

$$6y^{8}+y^{8}$$

Combine 6y^{8} and y^{8} to get 7y^{8}.

$$7y^{8}$$

Differentiate w.r.t. y

$$56y^{7}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 5 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{8}\times 3+\left(y^{2}\right)^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{8}\times 3+y^{8})$$

Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}y}(6y^{8}+y^{8})$$

Combine 6y^{8} and y^{8} to get 7y^{8}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(7y^{8})$$

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

$$8\times 7y^{8-1}$$

Multiply 8 times 7.

$$56y^{8-1}$$

Subtract 1 from 8.

$$56y^{7}$$