Solve 0.3x^3*12x^2/0.8x^9

$$\frac { 0.3 x ^ { 3 } \times 12 x ^ { 2 } } { 0.8 x ^ { 9 } }$$

Evaluate

$$\frac{9}{2x^{4}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{0.3x^{5}\times 12}{0.8x^{9}}$$

Cancel out x^{5} in both numerator and denominator.

$$\frac{0.3\times 12}{0.8x^{4}}$$

Multiply 0.3 and 12 to get 3.6.

$$\frac{3.6}{0.8x^{4}}$$

Differentiate w.r.t. x

$$-\frac{18}{x^{5}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{36x^{2}}{0.8\times 10}x^{3-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x^{2}}{2}x^{-6})$$

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}.

$$-6\times \frac{9x^{2}}{2}x^{-6-1}$$

Do the arithmetic.

$$\left(-27x^{2}\right)x^{-7}$$