Solve (y^4)^5+(y^5)^4

$$( y ^ { 4 } ) ^ { 5 } + ( y ^ { 5 } ) ^ { 4 }$$

Evaluate

$$2y^{20}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 5 to get 20.

$$y^{20}+\left(y^{5}\right)^{4}$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$y^{20}+y^{20}$$

Combine y^{20} and y^{20} to get 2y^{20}.

$$2y^{20}$$

Differentiate w.r.t. y

$$40y^{19}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 4 and 5 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{20}+\left(y^{5}\right)^{4})$$

To raise a power to another power, multiply the exponents. Multiply 5 and 4 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{20}+y^{20})$$

Combine y^{20} and y^{20} to get 2y^{20}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{20})$$

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

$$20\times 2y^{20-1}$$

Multiply 20 times 2.

$$40y^{20-1}$$

Subtract 1 from 20.

$$40y^{19}$$