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Derivative of:
Derivative of e^2*x
Derivative of cos(x)^2
Derivative of x^sin(x)
Derivative of sin(x^3)
Integral of d{x}:
x^2*5^x
Identical expressions
x^ two * five ^x
x squared multiply by 5 to the power of x
x to the power of two multiply by five to the power of x
x2*5x
x²*5^x
x to the power of 2*5 to the power of x
x^25^x
x25x

The derivative of the function/ x^2*5^x

Derivative of x^2*5^x

The solution

You have entered [src]

 2  x
x *5

$$5^{x} x^{2}$$

x^2*5^x

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{2}$ goes to $2 x$
$g{\left(x \right)} = 5^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
The result is: $5^{x} x^{2} \log{\left(5 \right)} + 2 \cdot 5^{x} x$
Now simplify:

$5^{x} x \left(x \log{\left(5 \right)} + 2\right)$

The answer is:

$5^{x} x \left(x \log{\left(5 \right)} + 2\right)$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     x    x  2       
2*x*5  + 5 *x *log(5)

$$5^{x} x^{2} \log{\left(5 \right)} + 2 \cdot 5^{x} x$$

Simplify

The second derivative [src]

 x /     2    2                \
5 *\2 + x *log (5) + 4*x*log(5)/

$$5^{x} \left(x^{2} \log{\left(5 \right)}^{2} + 4 x \log{\left(5 \right)} + 2\right)$$

Simplify

The third derivative [src]

 x /     2    2                \       
5 *\6 + x *log (5) + 6*x*log(5)/*log(5)

$$5^{x} \left(x^{2} \log{\left(5 \right)}^{2} + 6 x \log{\left(5 \right)} + 6\right) \log{\left(5 \right)}$$

Simplify

The graph

Derivative of x^2*5^x