Solve aquada^23^4-6a^2+12a-8-8a

$$a \quad a ^ { 2 } 3 ^ { 4 } - 6 a ^ { 2 } + 12 a - 8 - 8 a$$

Evaluate

$$81a^{3}-6a^{2}+4a-8$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$a^{3}\times 3^{4}-6a^{2}+12a-8-8a$$

Calculate 3 to the power of 4 and get 81.

$$a^{3}\times 81-6a^{2}+12a-8-8a$$

Combine 12a and -8a to get 4a.

$$a^{3}\times 81-6a^{2}+4a-8$$

Differentiate w.r.t. a

$$243a^{2}-12a+4$$

Steps Using Derivative Rule for Sum

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3}\times 3^{4}-6a^{2}+12a-8-8a)$$

Calculate 3 to the power of 4 and get 81.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3}\times 81-6a^{2}+12a-8-8a)$$

Combine 12a and -8a to get 4a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{3}\times 81-6a^{2}+4a-8)$$

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}.

$$3\times 81a^{3-1}+2\left(-6\right)a^{2-1}+4a^{1-1}$$

Multiply 3 times 81.

$$243a^{3-1}+2\left(-6\right)a^{2-1}+4a^{1-1}$$

Subtract 1 from 3.

$$243a^{2}+2\left(-6\right)a^{2-1}+4a^{1-1}$$

Multiply 2 times -6.

$$243a^{2}-12a^{2-1}+4a^{1-1}$$

Subtract 1 from 2.

$$243a^{2}-12a^{1}+4a^{1-1}$$

Subtract 1 from 1.

$$243a^{2}-12a^{1}+4a^{0}$$

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

$$243a^{2}-12a+4a^{0}$$

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

$$243a^{2}-12a+4\times 1$$

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

$$243a^{2}-12a+4$$