In product rule calculus, we use the multiplication rule of derivatives whilst two or greater features have become elevated. If we've functions f(x) and g(x), then the product rule states that: “ f(x) times the by-product of g(x) plus g(x) times the derivative of f(x)”
suppose that we've got two functions f(x) and g(x) that are differentiable. The by-product product rule components for those capabilities is as follows:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
other than the usage of formulation for manual calculations, use on-line product rule derivative calculator totally free to find derivative of two product capabilities.
Example :
Differentiate the following function w.r t x.
$$ h\left( x \right) = \left( {4{x^2} + 5x} \right)\left( {2 - 10x} \right) $$
Solution:
The given function is:
$$ h\left( x \right) = \left( {4{x^2} + 5x} \right)\left( {2 - 10x} \right) $$
As we know that the multiplication derivative rule is as follows:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Here we have separate functions according to the formula as follows:
$$ f(x) = \left( {4{x^2} + 5x} \right) $$
$$ g(x) = \left( {2 - 10x} \right) $$
Now, calculating the derivative of f(x) w.r.t x:
$$ \frac{d}{d x} f(x) $$
$$ = \frac{d}{d x} \left( 4x^2 + 5x \right) $$
$$ \frac{d}{d x} \left( 4x^2 + 5x \right) = \left( 8x + 5 \right) $$
$$ \frac{d}{d x} g(x) $$
$$ = \frac{d}{d x} \left( 2 - 10x \right) $$
$$ \frac{d}{d x} \left( 2 - 10x \right) = -10 $$
(For step-by-step calculation of derivative, click derivative calculator)
Now, according to the multiplication rule for derivatives:
$$ \frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)} $$
Putting derivatives in the formula to get the final answer.
$$ \left( 4x^2 + 5x \right) (-10) + \left( 2 - 10x \right) \left( 8x + 5 \right) $$
$$ -40x^2 - 50x + \left( 2 - 10x \right) \left( 8x + 5 \right) $$ after simplifying , we get:
$$ -40x^2 - 50x + (2)(8x) + (2)(5) - (10x)(8x) - (10x)(5) $$
$$ -40x^2 - 50x + 16x + 10 - 80x^2 - 50x $$
$$ -120x^2 - 100x + 10 $$
So we have:
$$ h\left( x \right) = \left( {4{x^2} + 5x} \right)\left( {2 - 10x} \right) = -120x^2 - 100x + 10 $$
Which is the required answer. Also, our free online product rule derivative calculator evaluates the given functions more accurately and instantly.
sure, you can do so. All you need to do is consider derivatives for each new function inside the expression and upload them to get the final solution.
The natural logarithm (ln) is defined handiest for x>0. this is why herbal log of zero is undefined. ln(0) = ∞
