Solve 23a^{5}-8a^3+12a^5-10a^5+11-5a^3

$$23 a ^ { 5 } - 8 a ^ { 3 } + 12 a ^ { 5 } - 10 a ^ { 5 } + 11 - 5 a ^ { 3 }$$

Evaluate

$$25a^{5}-13a^{3}+11$$

Solution Steps

Combine 23a^{5} and 12a^{5} to get 35a^{5}.

$$35a^{5}-8a^{3}-10a^{5}+11-5a^{3}$$

Combine 35a^{5} and -10a^{5} to get 25a^{5}.

$$25a^{5}-8a^{3}+11-5a^{3}$$

Combine -8a^{3} and -5a^{3} to get -13a^{3}.

$$25a^{5}-13a^{3}+11$$

Differentiate w.r.t. a

$$a^{2}\left(125a^{2}-39\right)$$

Steps Using Derivative Rule for Sum

Combine 23a^{5} and 12a^{5} to get 35a^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(35a^{5}-8a^{3}-10a^{5}+11-5a^{3})$$

Combine 35a^{5} and -10a^{5} to get 25a^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(25a^{5}-8a^{3}+11-5a^{3})$$

Combine -8a^{3} and -5a^{3} to get -13a^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(25a^{5}-13a^{3}+11)$$

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}.

$$5\times 25a^{5-1}+3\left(-13\right)a^{3-1}$$

Multiply 5 times 25.

$$125a^{5-1}+3\left(-13\right)a^{3-1}$$

Subtract 1 from 5.

$$125a^{4}+3\left(-13\right)a^{3-1}$$

Multiply 3 times -13.

$$125a^{4}-39a^{3-1}$$

Subtract 1 from 3.

$$125a^{4}-39a^{2}$$