Solve 2p^2/15pq*3p^2q/54

$$\frac { 2 p ^ { 2 } } { 15 p q } \times \frac { 3 p ^ { 2 } q } { 54 }$$

Evaluate

$$\frac{p^{3}}{135}$$

Solution Steps

Cancel out p in both numerator and denominator.

$$\frac{2p}{15q}\times \frac{3p^{2}q}{54}$$

Divide 3p^{2}q by 54 to get \frac{1}{18}p^{2}q.

$$\frac{2p}{15q}\times \frac{1}{18}p^{2}q$$

Multiply \frac{2p}{15q} times \frac{1}{18} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2p}{15q\times 18}p^{2}q$$

Cancel out 2 in both numerator and denominator.

$$\frac{p}{9\times 15q}p^{2}q$$

Express \frac{p}{9\times 15q}p^{2} as a single fraction.

$$\frac{pp^{2}}{9\times 15q}q$$

Express \frac{pp^{2}}{9\times 15q}q as a single fraction.

$$\frac{pp^{2}q}{9\times 15q}$$

Cancel out q in both numerator and denominator.

$$\frac{pp^{2}}{9\times 15}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{p^{3}}{9\times 15}$$

Multiply 9 and 15 to get 135.

$$\frac{p^{3}}{135}$$

Differentiate w.r.t. p

$$\frac{p^{2}}{45}$$

Steps Using Definition of a Derivative

Cancel out p in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{2p}{15q}\times \frac{3p^{2}q}{54})$$

Divide 3p^{2}q by 54 to get \frac{1}{18}p^{2}q.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{2p}{15q}\times \frac{1}{18}p^{2}q)$$

Multiply \frac{2p}{15q} times \frac{1}{18} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{2p}{15q\times 18}p^{2}q)$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p}{9\times 15q}p^{2}q)$$

Express \frac{p}{9\times 15q}p^{2} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{pp^{2}}{9\times 15q}q)$$

Express \frac{pp^{2}}{9\times 15q}q as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{pp^{2}q}{9\times 15q})$$

Cancel out q in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{pp^{2}}{9\times 15})$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p^{3}}{9\times 15})$$

Multiply 9 and 15 to get 135.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{p^{3}}{135})$$

The derivative of ax^{n} is nax^{n-1}.

$$3\times \frac{1}{135}p^{3-1}$$

Multiply 3 times \frac{1}{135}.

$$\frac{1}{45}p^{3-1}$$

Subtract 1 from 3.

$$\frac{1}{45}p^{2}$$