Solve (y^7)^2/y^20

$$\frac { ( y ^ { 7 } ) ^ { 2 } } { y ^ { 20 } }$$

Evaluate

$$\frac{1}{y^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(y^{7}\right)^{2}\times \frac{1}{y^{20}}$$

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

$$y^{7\times 2}y^{20\left(-1\right)}$$

Multiply 7 times 2.

$$y^{14}y^{20\left(-1\right)}$$

Multiply 20 times -1.

$$y^{14}y^{-20}$$

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

$$y^{14-20}$$

Add the exponents 14 and -20.

$$y^{-6}$$

Differentiate w.r.t. y

$$-\frac{6}{y^{7}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{14}}{y^{20}})$$

Rewrite y^{20} as y^{14}y^{6}. Cancel out y^{14} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{6}})$$

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).

$$-\left(y^{6}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{6})$$

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}.

$$-\left(y^{6}\right)^{-2}\times 6y^{6-1}$$

Simplify.

$$-6y^{5}\left(y^{6}\right)^{-2}$$