Solve 12x^5^-9x^3+3x^3div3{x}^2

$$12 { x }^{ { 5 }^{ } } -9 { x }^{ 3 } +3 { x }^{ 3 } \div 3 { x }^{ 2 }$$

Evaluate

$$\left(13x^{2}-9\right)x^{3}$$

Solution Steps

Calculate 5 to the power of 1 and get 5.

$$12x^{5}-9x^{3}+\frac{3x^{3}}{3}x^{2}$$

Cancel out 3 and 3.

$$12x^{5}-9x^{3}+x^{3}x^{2}$$

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

$$12x^{5}-9x^{3}+x^{5}$$

Combine 12x^{5} and x^{5} to get 13x^{5}.

$$13x^{5}-9x^{3}$$

Differentiate w.r.t. x

$$x^{2}\left(65x^{2}-27\right)$$

Steps Using Derivative Rule for Sum

Calculate 5 to the power of 1 and get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x^{5}-9x^{3}+\frac{3x^{3}}{3}x^{2})$$

Cancel out 3 and 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x^{5}-9x^{3}+x^{3}x^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x^{5}-9x^{3}+x^{5})$$

Combine 12x^{5} and x^{5} to get 13x^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(13x^{5}-9x^{3})$$

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}.

$$5\times 13x^{5-1}+3\left(-9\right)x^{3-1}$$

Multiply 5 times 13.

$$65x^{5-1}+3\left(-9\right)x^{3-1}$$

Subtract 1 from 5.

$$65x^{4}+3\left(-9\right)x^{3-1}$$

Multiply 3 times -9.

$$65x^{4}-27x^{3-1}$$

Subtract 1 from 3.

$$65x^{4}-27x^{2}$$