Solve frac{{a}^frac{3{4}}a^-1div4}{{a}^-frac{10{4}}}

$$\frac{ { a }^{ \frac{ 3 }{ 4 } } { a }^{ -1 \div 4 } }{ { a }^{ - \frac{ 10 }{ 4 } } }$$

Evaluate

$$a^{3}$$

Solution Steps

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

$$\frac{a^{\frac{3}{4}}a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}$$

To multiply powers of the same base, add their exponents. Add \frac{3}{4} and -\frac{1}{4} to get \frac{1}{2}.

$$\frac{a^{\frac{1}{2}}}{a^{-\frac{10}{4}}}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$\frac{a^{\frac{1}{2}}}{a^{-\frac{5}{2}}}$$

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

$$a^{3}$$

Differentiate w.r.t. a

$$3a^{2}$$

Steps Using Definition of a Derivative

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{\frac{3}{4}}a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}})$$

To multiply powers of the same base, add their exponents. Add \frac{3}{4} and -\frac{1}{4} to get \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{\frac{1}{2}}}{a^{-\frac{10}{4}}})$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{\frac{1}{2}}}{a^{-\frac{5}{2}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3})$$

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

$$3a^{3-1}$$

Subtract 1 from 3.

$$3a^{2}$$