Solve 9x^5*27x^2+45x^4*9x^3

$$9 { x }^{ 5 } \times 27 { x }^{ 2 } +45 { x }^{ 4 } \times 9 { x }^{ 3 }$$

Evaluate

$$648x^{7}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$9x^{7}\times 27+45x^{4}\times 9x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$9x^{7}\times 27+45x^{7}\times 9$$

Multiply 9 and 27 to get 243.

$$243x^{7}+45x^{7}\times 9$$

Multiply 45 and 9 to get 405.

$$243x^{7}+405x^{7}$$

Combine 243x^{7} and 405x^{7} to get 648x^{7}.

$$648x^{7}$$

Differentiate w.r.t. x

$$4536x^{6}$$

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}.

$$5\times 243x^{2}x^{5-1}+4\times 405x^{3}x^{4-1}$$

Multiply 5 times 9\times 27x^{2}.

$$1215x^{2}x^{5-1}+4\times 405x^{3}x^{4-1}$$

Subtract 1 from 5.

$$1215x^{2}x^{4}+4\times 405x^{3}x^{4-1}$$

Multiply 4 times 9\times 45x^{3}.

$$1215x^{2}x^{4}+1620x^{3}x^{4-1}$$

Subtract 1 from 4.

$$1215x^{2}x^{4}+1620x^{3}x^{3}$$