What is the derivative of ay(e^{by}) ?

The derivative of ay(e^{by}) is a(e^{by}+be^{by}y)

What is the first derivative of ay(e^{by}) ?

The first derivative of ay(e^{by}) is a(e^{by}+be^{by}y)

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derivative of

a y (e^{b y})

a (e^{b y} + b e^{b y} y)

Solution steps

\frac{d}{d y} (a y e^{b y})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= a \frac{d}{d y} (y e^{b y})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = e^{b y}

= a (\frac{d y}{d y} e^{b y} + \frac{d}{d y} (e^{b y}) y)

\frac{d y}{d y} = 1

\frac{d}{d y} (e^{b y}) = e^{b y} b

= a (1 \cdot e^{b y} + e^{b y} b y)

Simplify

= a (e^{b y} + b e^{b y} y)

ln (- x)

\int \frac{12}{x ^{- 1} + 3} d x

\frac{d y}{d x} = 3 + 12 x + 9 y + 36 x y

\frac{d}{d x} (\frac{1}{2 x + 9})

\int_{1}^{4} (2 x + 1) d x

What is the derivative of ay(e^{by}) ?
The derivative of ay(e^{by}) is a(e^{by}+be^{by}y)
What is the first derivative of ay(e^{by}) ?
The first derivative of ay(e^{by}) is a(e^{by}+be^{by}y)