# Use the Product Rule to Simplify the Expression

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Use the Product Rule to Simplify the Expression
Use the Product Rule to Simplify the Expression
We use the product rule to simplify the expression where expression is a
combination of different kind of variables, numbers and operations like addition,
subtraction, multiplication and division.Now, we discuss how product rule
simplify different kind of expressions:
Combination of algebraic and exponential: If expression is a combination of
algebraic and exponential function, then with the help of product rule we can
easily solve differentiation of that expression-
d (x.ex) = x. d (ex) + ex.d (x)
dx dx dx
= x.ex + ex.1
= ex(x+1)

Know More About Antiderivative of a x

Combination of algebraic and trigonometric: If expression is a combination of
algebraic and trigonometric function, then-
d (x2.sin x) = x2. d (sin x) + sinx.d (x2)
dx dx dx
= x2.(cos x) + sin x .(2x)
= x(x.cos x + 2.sin x)
Combination of algebraic and logarithm: If expression is a combination of
algebraic and logarithm function, then-
d (x.ln x) = x. d (ln x) + ln x.d (x)
dx dx dx
= x.(1) + ln x.(1)
x
= 1+ ln x
Combination of trigonometric: If expression is a combination of two
trigonometric functions, then-
d (sin x. cosx) = sin x. d (cos x) + cos x.d (sin x)

dx dx dx
= sin x.(-sin2x) + cos x.(cos x)
= cos2x - sin2x
= cos2x
Combination of trigonometric and exponential: If expression is a combination of
trigonometric and exponential function, then-
d (sin x.ex) = sin x. d (ex) + ex. d (sin x)
dx dx dx
= sin x. (ex) + ex.(cos x)
= ex (sinx + cos x)
These are different expressions which we simplify by product rule.

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