What is the derivative of 2^{5x}3^{4x^2} ?

The derivative of 2^{5x}3^{4x^2} is 5ln(2)*32^x*81^{x^2}+ln(3)*81^{x^2}*2^{3+5x}x

What is the first derivative of 2^{5x}3^{4x^2} ?

The first derivative of 2^{5x}3^{4x^2} is 5ln(2)*32^x*81^{x^2}+ln(3)*81^{x^2}*2^{3+5x}x

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\frac{d}{d x} (2^{5 x} 3^{4 x^{2}})

5 ln (2) \cdot 3 2^{x} \cdot 8 1^{x^{2}} + ln (3) \cdot 8 1^{x^{2}} \cdot 2^{3 + 5 x} x

Solution steps

\frac{d}{d x} (2^{5 x} \cdot 3^{4 x^{2}})

Simplify

2^{5 x} \cdot 3^{4 x^{2}} : 3 2^{x} \cdot 8 1^{x^{2}}

= \frac{d}{d x} (3 2^{x} \cdot 8 1^{x^{2}})

Apply the Product Rule:

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

f = 3 2^{x}, g = 8 1^{x^{2}}

= \frac{d}{d x} (3 2^{x}) 8 1^{x^{2}} + \frac{d}{d x} (8 1^{x^{2}}) 3 2^{x}

\frac{d}{d x} (3 2^{x}) = 5 ln (2) \cdot 3 2^{x}

\frac{d}{d x} (8 1^{x^{2}}) = 8 ln (3) \cdot 8 1^{x^{2}} x

= 5 ln (2) \cdot 3 2^{x} \cdot 8 1^{x^{2}} + 8 ln (3) \cdot 8 1^{x^{2}} x \cdot 3 2^{x}

8 ln (3) \cdot 8 1^{x^{2}} x \cdot 3 2^{x} = ln (3) \cdot 8 1^{x^{2}} \cdot 2^{3 + 5 x} x

= 5 ln (2) \cdot 3 2^{x} \cdot 8 1^{x^{2}} + ln (3) \cdot 8 1^{x^{2}} \cdot 2^{3 + 5 x} x

\frac{d y}{d x} = \frac{x - 4}{y + 5}

f (x) = \frac{2}{x}, (6, \frac{1}{3})

\int 15 ln (x + 5) d x

\int \frac{x ^{2} + 5 x - 1}{x ^{3}} d x

What is the derivative of 2^{5x}3^{4x^2} ?
The derivative of 2^{5x}3^{4x^2} is 5ln(2)*32^x*81^{x^2}+ln(3)*81^{x^2}*2^{3+5x}x
What is the first derivative of 2^{5x}3^{4x^2} ?
The first derivative of 2^{5x}3^{4x^2} is 5ln(2)*32^x*81^{x^2}+ln(3)*81^{x^2}*2^{3+5x}x