Answer to Question #154485 in Calculus for ice

Question

1.g(x)=2^5x 4^3x^2

2.Given 𝑒 𝑥 𝑠𝑖𝑛𝑦 + 𝑒 𝑦 𝑐𝑜𝑠𝑥 = 1 , find 𝑦′ .

3.y=e^x^3 3^e^x

Answer 1

$g^{\prime}(x) = (2^{5x})^{\prime} \cdot 4^{3x^2} + 2^{5x} \cdot (4^{3x^2})^{\prime} = 2^{5x} \cdot \ln2 \cdot 5 \cdot 4^{3x^2} + 2^{5x} \cdot 4^{3x^2} \cdot \ln4 \cdot 6 \cdot x = \newline 2^{5x} \cdot 4^{3x^2} \cdot (5 \cdot \ln2 + 6 \cdot x \cdot \ln4)$

$y^{\prime} = -\frac{F_x^{\prime}}{F_y^{\prime}} = -\frac{e^x \cdot siny - e^y sinx}{e^x \cdot cosy -+e^y cosx} = \frac{e^y sinx - e^x \cdot siny}{e^x \cdot cosy -+e^y cosx}$

$y^{\prime} = (e^{x^3})^{\prime} \cdot 3^{e^x} + e^{x^3} \cdot (3^{e^x})^{\prime} = e^{x^3} \cdot 3 \cdot x^2 \cdot 3^{e^x} + e^{x^3} \cdot 3^{e^x} \cdot \ln3 \cdot e^x = \newline e^{x^3} \cdot 3^{e^x} \cdot (3 \cdot x^2 + e^x \cdot \ln3)$