Solve frac{x^{30}*(x^2)^4}{x^-3divx^2}

$$\frac { x ^ { 30 } \cdot ( x ^ { 2 } ) ^ { 4 } } { x ^ { - 3 } \div x ^ { 2 } }$$

Evaluate

$$x^{43}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{x^{30}x^{8}}{\frac{x^{-3}}{x^{2}}}$$

To multiply powers of the same base, add their exponents. Add 30 and 8 to get 38.

$$\frac{x^{38}}{\frac{x^{-3}}{x^{2}}}$$

Rewrite x^{2} as x^{-3}x^{5}. Cancel out x^{-3} in both numerator and denominator.

$$\frac{x^{38}}{\frac{1}{x^{5}}}$$

Divide x^{38} by \frac{1}{x^{5}} by multiplying x^{38} by the reciprocal of \frac{1}{x^{5}}.

$$x^{38}x^{5}$$

To multiply powers of the same base, add their exponents. Add 38 and 5 to get 43.

$$x^{43}$$

Differentiate w.r.t. x

$$43x^{42}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{30}x^{8}}{\frac{x^{-3}}{x^{2}}})$$

To multiply powers of the same base, add their exponents. Add 30 and 8 to get 38.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{38}}{\frac{x^{-3}}{x^{2}}})$$

Rewrite x^{2} as x^{-3}x^{5}. Cancel out x^{-3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{38}}{\frac{1}{x^{5}}})$$

Divide x^{38} by \frac{1}{x^{5}} by multiplying x^{38} by the reciprocal of \frac{1}{x^{5}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{38}x^{5})$$

To multiply powers of the same base, add their exponents. Add 38 and 5 to get 43.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{43})$$

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

$$43x^{43-1}$$

Subtract 1 from 43.

$$43x^{42}$$