Solve (y^-4)^7

$$( y ^ { - 4 } ) ^ { 7 }$$

Evaluate

$$\frac{1}{y^{28}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\left(y^{-4}\right)^{7}$$

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

$$y^{-4\times 7}$$

Multiply -4 times 7.

$$\frac{1}{y^{28}}$$

Differentiate w.r.t. y

$$-\frac{28}{y^{29}}$$

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).

$$7\left(y^{-4}\right)^{7-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{-4})$$

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}.

$$7\left(y^{-4}\right)^{6}\left(-4\right)y^{-4-1}$$

Simplify.

$$-28y^{-5}\left(y^{-4}\right)^{6}$$