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If g(x)=(x^5+6)^18(7x^4+9)^28 , find g'(x). g'(x)=□

If g(x)=(x^5+6)^18(7x^4+9)^28 , find g'(x). g'(x)=□

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28(7x4+9)27∗28x3∗(x5+6)18+18(x5+6)17∗5x4∗(7x4+9)28 28(7x^{4}+9)^{27}*28x^{3}*(x^{5}+6)^{18}+18(x^{5}+6)^{17}*5x^{4}*(7x^{4}+9)^{28}
1 Recognize that g(x) is a product of two functions, so we will use the Product Rule for differentiation. The Product Rule states that (f(x)g(x))′=f′(x)g(x)+g′(x)f(x)(f(x)g(x))'=f'(x)g(x)+g'(x)f(x)
2 Let f(x)=(x5+6)18f(x)=(x^{5}+6)^{18} and g(x)=(7x4+9)28g(x)=(7x^{4}+9)^{28}
3 Differentiate f(x) using the Power Rule and the Chain Rule. The Power Rule states that ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}, and the Chain Rule states that ddx(un)=nun−1dudx\frac{d}{dx}(u^n) = nu^{n-1}\frac{du}{dx}
4 The derivative of f(x) is f′(x)=18(x5+6)17⋅5x4f'(x)=18(x^{5}+6)^{17} \cdot 5x^{4}
5 Differentiate g(x) using the Power Rule and the Chain Rule
6 The derivative of g(x) is g′(x)=28(7x4+9)27⋅28x3g'(x)=28(7x^{4}+9)^{27} \cdot 28x^{3}
7 Apply the Product Rule to find g′(x)g'(x):
g′(x)=18(x5+6)17⋅5x4⋅(7x4+9)28+28(7x4+9)27⋅28x3⋅(x5+6)18g'(x) = 18(x^{5}+6)^{17} \cdot 5x^{4} \cdot (7x^{4}+9)^{28} + 28(7x^{4}+9)^{27} \cdot 28x^{3} \cdot (x^{5}+6)^{18}
8 The final answer is the sum of the two terms obtained from the Product Rule.
So, the derivative of the function g(x) is 28(7x4+9)27∗28x3∗(x5+6)18+18(x5+6)17∗5x4∗(7x4+9)2828(7x^{4}+9)^{27}*28x^{3}*(x^{5}+6)^{18}+18(x^{5}+6)^{17}*5x^{4}*(7x^{4}+9)^{28}
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