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

The derivative of 3*4^{x^2+2} is 3ln(2)*2^{2x^2+6}x

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

The first derivative of 3*4^{x^2+2} is 3ln(2)*2^{2x^2+6}x

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

3 ln (2) \cdot 2^{2 x^{2} + 6} x

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

4^{x^{2} + 2} = e^{(x^{2} + 2) l n (4)}

= 3 \frac{d}{d x} (e^{(x^{2} + 2) l n (4)})

Apply the chain rule:

e^{(x^{2} + 2) l n (4)} \frac{d}{d x} ((x^{2} + 2) ln (4))

= e^{(x^{2} + 2) l n (4)} \frac{d}{d x} ((x^{2} + 2) ln (4))

\frac{d}{d x} ((x^{2} + 2) ln (4)) = 4 ln (2) x

= 3 e^{(x^{2} + 2) l n (4)} \cdot 4 ln (2) x

Simplify

3 e^{(x^{2} + 2) l n (4)} \cdot 4 ln (2) x : 3 ln (2) \cdot 2^{2 x^{2} + 6} x

= 3 ln (2) \cdot 2^{2 x^{2} + 6} x

What is the derivative of 3*4^{x^2+2} ?
The derivative of 3*4^{x^2+2} is 3ln(2)*2^{2x^2+6}x
What is the first derivative of 3*4^{x^2+2} ?
The first derivative of 3*4^{x^2+2} is 3ln(2)*2^{2x^2+6}x