Solve 2x^2*1/3x^3+21x^7/3x^2=

$$2 x ^ { 2 } \cdot \frac { 1 } { 3 } x ^ { 3 } + \frac { 21 x ^ { 7 } } { 3 x ^ { 2 } } =$$

Evaluate

$$\frac{23x^{5}}{3}$$

Solution Steps

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

$$2x^{5}\times \frac{1}{3}+\frac{21x^{7}}{3x^{2}}$$

Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$\frac{2}{3}x^{5}+\frac{21x^{7}}{3x^{2}}$$

Cancel out 3x^{2} in both numerator and denominator.

$$\frac{2}{3}x^{5}+7x^{5}$$

Combine \frac{2}{3}x^{5} and 7x^{5} to get \frac{23}{3}x^{5}.

$$\frac{23}{3}x^{5}$$

Differentiate w.r.t. x

$$\frac{115x^{4}}{3}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{5}\times \frac{1}{3}+\frac{21x^{7}}{3x^{2}})$$

Multiply 2 and \frac{1}{3} to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}x^{5}+\frac{21x^{7}}{3x^{2}})$$

Cancel out 3x^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{3}x^{5}+7x^{5})$$

Combine \frac{2}{3}x^{5} and 7x^{5} to get \frac{23}{3}x^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{23}{3}x^{5})$$

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

$$5\times \frac{23}{3}x^{5-1}$$

Multiply 5 times \frac{23}{3}.

$$\frac{115}{3}x^{5-1}$$

Subtract 1 from 5.

$$\frac{115}{3}x^{4}$$