Solve sqrt{81}a^3-5asqrt{a}+3/asqrt{4a^5}

$$\sqrt { 81 } a ^ { 3 } - 5 a \sqrt { a } + \frac { 3 } { a } \sqrt { 4 a ^ { 5 } }$$

Differentiate w.r.t. a

$$27a^{2}+\frac{3\sqrt{a}}{2}$$

Steps Using Derivative Rule for Sum

Calculate the square root of 81 and get 9.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a^{3}-5a\sqrt{a}+\frac{3}{a}\sqrt{4a^{5}})$$

Express \frac{3}{a}\sqrt{4a^{5}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a^{3}-5a\sqrt{a}+\frac{3\sqrt{4a^{5}}}{a})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9a^{3} times \frac{a}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9a^{3}a}{a}-5a\sqrt{a}+\frac{3\sqrt{4a^{5}}}{a})$$

Since \frac{9a^{3}a}{a} and \frac{3\sqrt{4a^{5}}}{a} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9a^{3}a+3\sqrt{4a^{5}}}{a}-5a\sqrt{a})$$

Do the multiplications in 9a^{3}a+3\sqrt{4a^{5}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{9a^{4}+6a^{\frac{5}{2}}}{a}-5a\sqrt{a})$$

Factor the expressions that are not already factored in \frac{9a^{4}+6a^{\frac{5}{2}}}{a}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3\left(3a^{\frac{3}{2}}+2\right)a^{\frac{5}{2}}}{a}-5a\sqrt{a})$$

Cancel out a in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(3a^{\frac{3}{2}}\left(3a^{\frac{3}{2}}+2\right)-5a\sqrt{a})$$

Expand the expression.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a^{3}+6a^{\frac{3}{2}}-5a\sqrt{a})$$

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}.

$$3\times 9a^{3-1}+\frac{3}{2}\times 6a^{\frac{3}{2}-1}+\left(-5\sqrt{a}\right)a^{1-1}$$

Multiply 3 times 9.

$$27a^{3-1}+\frac{3}{2}\times 6a^{\frac{3}{2}-1}+\left(-5\sqrt{a}\right)a^{1-1}$$

Subtract 1 from 3.

$$27a^{2}+\frac{3}{2}\times 6a^{\frac{3}{2}-1}+\left(-5\sqrt{a}\right)a^{1-1}$$

Multiply \frac{3}{2} times 6.

$$27a^{2}+9a^{\frac{3}{2}-1}+\left(-5\sqrt{a}\right)a^{1-1}$$

Subtract 1 from \frac{3}{2}.

$$27a^{2}+9\sqrt{a}+\left(-5\sqrt{a}\right)a^{1-1}$$

Subtract 1 from 1.

$$27a^{2}+9\sqrt{a}+\left(-5\sqrt{a}\right)a^{0}$$

For any term t except 0, t^{0}=1.

$$27a^{2}+9\sqrt{a}+\left(-5\sqrt{a}\right)\times 1$$

For any term t, t\times 1=t and 1t=t.

$$27a^{2}+9\sqrt{a}-5\sqrt{a}$$

Evaluate

$$9a^{3}+a^{\frac{3}{2}}$$