Solve frac{36x^frac{3{4}}(x+2)^frac{7{2}}}{9x^-frac{1{4}}(x+2)^frac{1{2}}}

$$\frac { 36 x ^ { \frac { 3 } { 4 } } ( x + 2 ) ^ { \frac { 7 } { 2 } } } { 9 x ^ { - \frac { 1 } { 4 } } ( x + 2 ) ^ { \frac { 1 } { 2 } } }$$

Differentiate w.r.t. x

$$8\left(2x+1\right)\left(x+2\right)^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{36\left(x+2\right)^{\frac{7}{2}}}{9\sqrt{x+2}}x^{\frac{3}{4}-\left(-\frac{1}{4}\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4\left(x+2\right)^{3}x^{1})$$

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}.

$$4\left(x+2\right)^{3}x^{1-1}$$

Do the arithmetic.

$$4\left(x+2\right)^{3}x^{0}$$

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

$$4\left(x+2\right)^{3}\times 1$$

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

$$4\left(x+2\right)^{3}$$

Evaluate

$$4x\left(x+2\right)^{3}$$