Solve y^2y^0/y^7

$$\frac{ { y }^{ 2 } { y }^{ 0 } }{ { y }^{ 7 } }$$

Evaluate

$$\frac{1}{y^{5}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 0 to get 2.

$$\frac{y^{2}}{y^{7}}$$

Rewrite y^{7} as y^{2}y^{5}. Cancel out y^{2} in both numerator and denominator.

$$\frac{1}{y^{5}}$$

Differentiate w.r.t. y

$$-\frac{5}{y^{6}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 2 and 0 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{2}}{y^{7}})$$

Rewrite y^{7} as y^{2}y^{5}. Cancel out y^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{5}})$$

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).

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

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}.

$$-\left(y^{5}\right)^{-2}\times 5y^{5-1}$$

Simplify.

$$-5y^{4}\left(y^{5}\right)^{-2}$$