Solve (y^4)^5

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

Evaluate

$$y^{20}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

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

To raise a power to another power, multiply the exponents.

$$y^{4\times 5}$$

Multiply 4 times 5.

$$y^{20}$$

Differentiate w.r.t. y

$$20y^{19}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$5\left(y^{4}\right)^{5-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{4})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$5\left(y^{4}\right)^{4}\times 4y^{4-1}$$

Simplify.

$$20y^{3}\left(y^{4}\right)^{4}$$