Solve [(y^3)]^5/[(y^6)]^4

$$\frac { [ ( y ^ { 3 } ) ] ^ { 5 } } { [ ( y ^ { 6 } ) ] ^ { 4 } }$$

Evaluate

$$\frac{1}{y^{9}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{y^{15}}{\left(y^{6}\right)^{4}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and 4 to get 24.

$$\frac{y^{15}}{y^{24}}$$

Rewrite y^{24} as y^{15}y^{9}. Cancel out y^{15} in both numerator and denominator.

$$\frac{1}{y^{9}}$$

Differentiate w.r.t. y

$$-\frac{9}{y^{10}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{15}}{\left(y^{6}\right)^{4}})$$

To raise a power to another power, multiply the exponents. Multiply 6 and 4 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{y^{15}}{y^{24}})$$

Rewrite y^{24} as y^{15}y^{9}. Cancel out y^{15} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{9}})$$

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).

$$-\left(y^{9}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{9})$$

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}.

$$-\left(y^{9}\right)^{-2}\times 9y^{9-1}$$

Simplify.

$$-9y^{8}\left(y^{9}\right)^{-2}$$