Solve quad(y^2)^3divy^63

$$\quad ( y ^ { 2 } ) ^ { 3 } \div y ^ { 63 }$$

Evaluate

$$\frac{1}{y^{57}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(y^{2}\right)^{3}\times \frac{1}{y^{63}}$$

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

$$y^{2\times 3}y^{63\left(-1\right)}$$

Multiply 2 times 3.

$$y^{6}y^{63\left(-1\right)}$$

Multiply 63 times -1.

$$y^{6}y^{-63}$$

To multiply powers of the same base, add their exponents.

$$y^{6-63}$$

Add the exponents 6 and -63.

$$y^{-57}$$

Differentiate w.r.t. y

$$-\frac{57}{y^{58}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{6}}{y^{63}})$$

Rewrite y^{63} as y^{6}y^{57}. Cancel out y^{6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{57}})$$

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).

$$-\left(y^{57}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{57})$$

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}.

$$-\left(y^{57}\right)^{-2}\times 57y^{57-1}$$

Simplify.

$$-57y^{56}\left(y^{57}\right)^{-2}$$