Solve T(x)=-8x^4+36x^6-54x^3+27

$$T ( x ) = - 8 x ^ { 4 } + 36 x ^ { 6 } - 54 x ^ { 3 } + 27$$

Differentiate w.r.t. x

$$2x^{2}\left(108x^{3}-16x-81\right)$$

Steps Using Derivative Rule for Sum

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}.

$$4\left(-8\right)x^{4-1}+6\times 36x^{6-1}+3\left(-54\right)x^{3-1}$$

Multiply 4 times -8.

$$-32x^{4-1}+6\times 36x^{6-1}+3\left(-54\right)x^{3-1}$$

Subtract 1 from 4.

$$-32x^{3}+6\times 36x^{6-1}+3\left(-54\right)x^{3-1}$$

Multiply 6 times 36.

$$-32x^{3}+216x^{6-1}+3\left(-54\right)x^{3-1}$$

Subtract 1 from 6.

$$-32x^{3}+216x^{5}+3\left(-54\right)x^{3-1}$$

Multiply 6 times 36.

$$-32x^{3}+216x^{5}-162x^{3-1}$$

Subtract 1 from 3.

$$-32x^{3}+216x^{5}-162x^{2}$$

Evaluate

$$36x^{6}-8x^{4}-54x^{3}+27$$